Assignment 5 Due: 11:59pm on Wednesday, March 5, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 6.13 A hand presses down on the book in the figure. Part A Is the normal force of the table on the book larger than, smaller than, or equal to ? ANSWER: Correct mg Equal to Larger than Smaller than mg mg mg Problem 6.2 The three ropes in the figure are tied to a small, very light ring. Two of these ropes are anchored to walls at right angles with the tensions shown in the figure. Part A What is the magnitude of the tension in the third rope? Express your answer using two significant figures. ANSWER: Correct Part B What is the direction of the tension in the third rope? Express your answer using two significant figures. T  3 T3 = 94 N T  3 Typesetting math: 100% ANSWER: Correct The Normal Force When an object rests on a surface, there is always a force perpendicular to the surface; we call this the normal force, denoted by . The two questions to the right will explore the normal force. Part A A man attempts to pick up his suitcase of weight by pulling straight up on the handle. However, he is unable to lift the suitcase from the floor. Which statement about the magnitude of the normal force acting on the suitcase is true during the time that the man pulls upward on the suitcase? Hint 1. How to approach this problem First, identify the forces that act on the suitcase and draw a free-body diagram. Then use the fact that the suitcase is in equilibrium, , to examine how the forces acting on the suitcase relate to each other. Hint 2. Identify the correct free-body diagram Which of the figures represents the free-body diagram of the suitcase while the man is pulling on the handle with a force of magnitude ? = 58   below horizontal n ws n F = 0 fpull Typesetting math: 100% ANSWER: ANSWER: Correct Part B A B C D The magnitude of the normal force is equal to the magnitude of the weight of the suitcase. The magnitude of the normal force is equal to the magnitude of the weight of the suitcase minus the magnitude of the force of the pull. The magnitude of the normal force is equal to the sum of the magnitude of the force of the pull and the magnitude of the suitcase’s weight. The magnitude of the normal force is greater than the magnitude of the weight of the suitcase. Typesetting math: 100% Now assume that the man of weight is tired and decides to sit on his suitcase. Which statement about the magnitude of the normal force acting on the suitcase is true during the time that the man is sitting on the suitcase? Hint 1. Identify the correct free-body diagram. Which of the figures represents the free-body diagram while the man is sitting atop the suitcase? Here the vector labeled is a force that has the same magnitude as the man’s weight. ANSWER: wm n wm Typesetting math: 100% ANSWER: Correct Recognize that the normal force acting on an object is not always equal to the weight of that object. This is an important point to understand. Problem 6.5 A construction worker with a weight of 880 stands on a roof that is sloped at 18 . Part A What is the magnitude of the normal force of the roof on the worker? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct A B C D The magnitude of the normal force is equal to the magnitude of the suitcase’s weight. The magnitude of the normal force is equal to the magnitude of the suitcase’s weight minus the magnitude of the man’s weight. The magnitude of the normal force is equal to the sum of the magnitude of the man’s weight and the magnitude of the suitcase’s weight. The magnitude of the normal force is less than the magnitude of the suitcase’s weight. N  n = 840 N Typesetting math: 100% Problem 6.6 In each of the two free-body diagrams, the forces are acting on a 3.0 object. Part A For diagram , find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B For diagram the part A, find the value of the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: kg ax x ax = -0.67 m s2 ay, y Typesetting math: 100% Correct Part C For diagram , find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D For diagram the part C, find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: ay = 0 m s2 ax x ax = 0.67 m s2 ay y Typesetting math: 100% Correct Problem 6.7 In each of the two free-body diagrams, the forces are acting on a 3.0 object. Part A Find the value of , the component of the acceleration in diagram (a). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct ay = 0 m s2 kg ax x ax = 0.99 m s2 Typesetting math: 100% Part B Find the value of , the component of the acceleration in diagram (a). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C Find the value of , the component of the acceleration in diagram (b). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D Find the value of , the component of the acceleration in diagram (b). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct ay y ay = 0 m s2 ax x ax = -0.18 m s2 ay y ay = 0 m s2 Typesetting math: 100% Problem 6.10 A horizontal rope is tied to a 53.0 box on frictionless ice. What is the tension in the rope if: Part A The box is at rest? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part B The box moves at a steady = 4.80 ? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part C The box = 4.80 and = 4.60 ? Express your answer to three significant figures and include the appropriate units. ANSWER: kg T = 0 N vx m/s T = 0 N vx m/s ax m/s2 Typesetting math: 100% Correct Problem 6.14 It takes the elevator in a skyscraper 4.5 to reach its cruising speed of 11 . A 60 passenger gets aboard on the ground floor. Part A What is the passenger’s weight before the elevator starts moving? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the passenger’s weight while the elevator is speeding up? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the passenger’s weight after the elevator reaches its cruising speed? T = 244 N s m/s kg w = 590 N w = 730 N Typesetting math: 100% Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Block on an Incline A block lies on a plane raised an angle from the horizontal. Three forces act upon the block: , the force of gravity; , the normal force; and , the force of friction. The coefficient of friction is large enough to prevent the block from sliding . Part A Consider coordinate system a, with the x axis along the plane. Which forces lie along the axes? ANSWER: w = 590 N  F  w F n F  f Typesetting math: 100% Correct Part B Which forces lie along the axes of the coordinate system b, in which the y axis is vertical? ANSWER: Correct only only only and and and and and F  f F  n F  w F  f F  n F  f F  w F  n F w F  f F  n F w only only only and and and and and F  f F  n F  w F  f F  n F  f F  w F  n F w F  f F  n F w Typesetting math: 100% Usually the best advice is to choose coordinate system so that the acceleration of the system is directly along one of the coordinate axes. If the system isn’t accelerating, then you are better off choosing the coordinate system with the most vectors along the coordinate axes. But now you are going to ignore that advice. You will find the normal force, , using vertical coordinate system b. In these coordinates you will find the magnitude appearing in both the x and y equations, each multiplied by a trigonometric function. Part C Because the block is not moving, the sum of the y components of the forces acting on the block must be zero. Find an expression for the sum of the y components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables , , , and . Hint 1. Find the y component of Write an expression for , the y component of the force , using coordinate system b. Express your answer in terms of and . Hint 1. Some geometry help – a useful angle The smaller angle between and the y-axis is also , as shown in the figure. ANSWER: F  n Fn Fn Ff Fw  F n Fny F  n Fn  F  n  Typesetting math: 100% Hint 2. Find the y component of Write an expression for , the y component of the force , using coordinate system b. Express your answer in terms of and . Hint 1. Some geometry help – a useful angle The smaller angle between and the x-axis is also , as shown in the figure. ANSWER: ANSWER: Fny = Fncos() F f Ffy F f Ff  F  f  Ffy = Ffsin() Fy = 0 = Fncos() + Ffsin() − Fw Typesetting math: 100% Correct Part D Because the block is not moving, the sum of the x components of the forces acting on the block must be zero. Find an expression for the sum of the x components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables , , , and . Hint 1. Find the x component of Write an expression for , the x component of the force , using coordinate system b. Express your answer in terms of and . ANSWER: ANSWER: Correct Part E To find the magnitude of the normal force, you must express in terms of since is an unknown. Using the equations you found in the two previous parts, find an expression for involving and but not . Hint 1. How to approach the problem From your answers to the previous two parts you should have two force equations ( and ). Combine these equations to eliminate . The key is to multiply the Fn Ff Fw  F n Fnx F  n Fn  Fnx = −Fnsin() Fx = 0 = −Fnsin() + Ffcos() Fn Fw Ff Fn Fw  Ff Typesetting math: 100% Fy = 0 Fx = 0 Ff equation for the y components by and the equation for the x components by , then add or subtract the two equations to eliminate the term . An alternative motivation for the algebra is to eliminate the trig functions in front of by using the trig identity . At the very least this would result in an equation that is simple to solve for . ANSWER: Correct Congratulations on working this through. Now realize that in coordinate system a, which is aligned with the plane, the y-coordinate equation is , which leads immediately to the result obtained here for . CONCLUSION: A thoughtful examination of which coordinate system to choose can save a lot of algebra. Contact Forces Introduced Learning Goal: To introduce contact forces (normal and friction forces) and to understand that, except for friction forces under certain circumstances, these forces must be determined from: net Force = ma. Two solid objects cannot occupy the same space at the same time. Indeed, when the objects touch, they exert repulsive normal forces on each other, as well as frictional forces that resist their slipping relative to each other. These contact forces arise from a complex interplay between the electrostatic forces between the electrons and ions in the objects and the laws of quantum mechanics. As two surfaces are pushed together these forces increase exponentially over an atomic distance scale, easily becoming strong enough to distort the bulk material in the objects if they approach too close. In everyday experience, contact forces are limited by the deformation or acceleration of the objects, rather than by the fundamental interatomic forces. Hence, we can conclude the following: The magnitude of contact forces is determined by , that is, by the other forces on, and acceleration of, the contacting bodies. The only exception is that the frictional forces cannot exceed (although they can be smaller than this or even zero). Normal and friction forces Two types of contact forces operate in typical mechanics problems, the normal and frictional forces, usually designated by and (or , or something similar) respectively. These are the components of the overall contact force: perpendicular to and parallel to the plane of contact. Kinetic friction when surfaces slide cos  sin  Ff cos() sin() Fn sin2() + cos2 () = 1 Fn Fn = Fwcos() Fy = Fn − FW cos() = 0 Fn F = ma μn n f Ffric n f Typesetting math: 100% When one surface is sliding past the other, experiments show three things about the friction force (denoted ): The frictional force opposes the relative motion at the 1. point of contact, 2. is proportional to the normal force, and 3. the ratio of the magnitude of the frictional force to that of the normal force is fairly constant over a wide range of speeds. The constant of proportionality is called the coefficient of kinetic friction, often designated . As long as the sliding continues, the frictional force is then (valid when the surfaces slide by each other). Static friction when surfaces don’t slide When there is no relative motion of the surfaces, the frictional force can assume any value from zero up to a maximum , where is the coefficient of static friction. Invariably, is larger than , in agreement with the observation that when a force is large enough that something breaks loose and starts to slide, it often accelerates. The frictional force for surfaces with no relative motion is therefore (valid when the contacting surfaces have no relative motion). The actual magnitude and direction of the static friction force are such that it (together with other forces on the object) causes the object to remain motionless with respect to the contacting surface as long as the static friction force required does not exceed . The equation is valid only when the surfaces are on the verge of sliding. Part A When two objects slide by one another, which of the following statements about the force of friction between them, is true? ANSWER: Correct Part B fk fk μk fk = μkn μsn μs μs μk fs ! μsn μsn fs = μsn The frictional force is always equal to . The frictional force is always less than . The frictional force is determined by other forces on the objects so it can be either equal to or less than . μkn μkn μkn Typesetting math: 100% When two objects are in contact with no relative motion, which of the following statements about the frictional force between them, is true? ANSWER: Correct For static friction, the actual magnitude and direction of the friction force are such that it, together with any other forces present, will cause the object to have the observed acceleration. The magnitude of the force cannot exceed . If the magnitude of static friction needed to keep acceleration equal to zero exceeds , then the object will slide subject to the resistance of kinetic friction. Do not automatically assume that unless you are considering a situation in which the magnitude of the static friction force is as large as possible (i.e., when determining at what point an object will just begin to slip). Whether the actual magnitude of the friction force is 0, less than , or equal to depends on the magnitude of the other forces (if any) as well as the acceleration of the object through . Part C When a board with a box on it is slowly tilted to larger and larger angle, common experience shows that the box will at some point “break loose” and start to accelerate down the board. The box begins to slide once the component of gravity acting parallel to the board just begins to exceeds the maximum force of static friction. Which of the following is the most general explanation for why the box accelerates down the board? ANSWER: The frictional force is always equal to . The frictional force is always less than . The frictional force is determined by other forces on the objects so it can be either equal to or less than . μsn μsn μsn μsn μsn fs = μsn μsn μsn F = ma Fg The force of kinetic friction is smaller than that of maximum static friction, but remains the same. Once the box is moving, is smaller than the force of maximum static friction but larger than the force of kinetic friction. Once the box is moving, is larger than the force of maximum static friction. When the box is stationary, equals the force of static friction, but once the box starts moving, the sliding reduces the normal force, which in turn reduces the friction. Fg Fg Fg Fg Typesetting math: 100% Correct At the point when the box finally does “break loose,” you know that the component of the box’s weight that is parallel to the board just exceeds (i.e., this component of gravitational force on the box has just reached a magnitude such that the force of static friction, which has a maximum value of , can no longer oppose it.) For the box to then accelerate, there must be a net force on the box along the board. Thus, the component of the box’s weight parallel to the board must be greater than the force of kinetic friction. Therefore the force of kinetic friction must be less than the force of static friction which implies , as expected. Part D Consider a problem in which a car of mass is on a road tilted at an angle . The normal force Select the best answer. ANSWER: Correct The key point is that contact forces must be determined from Newton’s equation. In the problem described above, there is not enough information given to determine the normal force (e.g., the acceleration is unknown). Each of the answer options is valid under some conditions ( , the car is sliding down an icy incline, or the car is going around a banked turn), but in fact none is likely to be correct if there are other forces on the car or if the car is accelerating. Do not memorize values for the normal force valid in different problems–you must determine from . Problem 6.17 Bonnie and Clyde are sliding a 323 bank safe across the floor to their getaway car. The safe slides with a constant speed if Clyde pushes from behind with 375 of force while Bonnie pulls forward on a rope with 335 of force. μsn μsn μkn μsn μk < μs M  is found using n = Mg n = Mg cos() n = Mg cos() F  = Ma  = 0 n F = ma kg N N Typesetting math: 100% Part A What is the safe's coefficient of kinetic friction on the bank floor? ANSWER: Correct Problem 6.19 A crate is placed on a horizontal conveyor belt. The materials are such that and . Part A Draw a free-body diagram showing all the forces on the crate if the conveyer belt runs at constant speed. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: 0.224 10 kg μs = 0.5 μk = 0.3 Typesetting math: 100% Correct Part B Draw a free-body diagram showing all the forces on the crate if the conveyer belt is speeding up. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: Typesetting math: 100% Correct Part C What is the maximum acceleration the belt can have without the crate slipping? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct amax = 4.9 m s2 Typesetting math: 100% Problem 6.28 A 1100 steel beam is supported by two ropes. Part A What is the tension in rope 1? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the tension in rope 2? Express your answer to two significant figures and include the appropriate units. ANSWER: kg T1 = 7000 N Typesetting math: 100% Correct Problem 6.35 The position of a 1.4 mass is given by , where is in seconds. Part A What is the net horizontal force on the mass at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the net horizontal force on the mass at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 6.39 T2 = 4800 N kg x = (2t3 − 3t2 )m t t = 0 s F = -8.4 N t = 1 s F = 8.4 N Typesetting math: 100% A rifle with a barrel length of 61 fires a 8 bullet with a horizontal speed of 400 . The bullet strikes a block of wood and penetrates to a depth of 11 . Part A What resistive force (assumed to be constant) does the wood exert on the bullet? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B How long does it take the bullet to come to rest after entering the wood? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 6.45 You and your friend Peter are putting new shingles on a roof pitched at 21 . You're sitting on the very top of the roof when Peter, who is at the edge of the roof directly below you, 5.0 away, asks you for the box of nails. Rather than carry the 2.0 box of nails down to Peter, you decide to give the box a push and have it slide down to him. Part A If the coefficient of kinetic friction between the box and the roof is 0.55, with what speed should you push the box to have it gently come to rest right at the edge of the roof? Express your answer to two significant figures and include the appropriate units. cm g m/s cm fk = 5800 N = 5.5×10−4 t s  m kg Typesetting math: 100% ANSWER: Correct Problem 6.54 The 2.0 wood box in the figure slides down a vertical wood wall while you push on it at a 45 angle. Part A What magnitude of force should you apply to cause the box to slide down at a constant speed? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct v = 3.9 ms kg  F = 23 N Typesetting math: 100% Score Summary: Your score on this assignment is 98.8%. You received 114.57 out of a possible total of 116 points. Typesetting math: 100%

Assignment 5 Due: 11:59pm on Wednesday, March 5, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 6.13 A hand presses down on the book in the figure. Part A Is the normal force of the table on the book larger than, smaller than, or equal to ? ANSWER: Correct mg Equal to Larger than Smaller than mg mg mg Problem 6.2 The three ropes in the figure are tied to a small, very light ring. Two of these ropes are anchored to walls at right angles with the tensions shown in the figure. Part A What is the magnitude of the tension in the third rope? Express your answer using two significant figures. ANSWER: Correct Part B What is the direction of the tension in the third rope? Express your answer using two significant figures. T  3 T3 = 94 N T  3 Typesetting math: 100% ANSWER: Correct The Normal Force When an object rests on a surface, there is always a force perpendicular to the surface; we call this the normal force, denoted by . The two questions to the right will explore the normal force. Part A A man attempts to pick up his suitcase of weight by pulling straight up on the handle. However, he is unable to lift the suitcase from the floor. Which statement about the magnitude of the normal force acting on the suitcase is true during the time that the man pulls upward on the suitcase? Hint 1. How to approach this problem First, identify the forces that act on the suitcase and draw a free-body diagram. Then use the fact that the suitcase is in equilibrium, , to examine how the forces acting on the suitcase relate to each other. Hint 2. Identify the correct free-body diagram Which of the figures represents the free-body diagram of the suitcase while the man is pulling on the handle with a force of magnitude ? = 58   below horizontal n ws n F = 0 fpull Typesetting math: 100% ANSWER: ANSWER: Correct Part B A B C D The magnitude of the normal force is equal to the magnitude of the weight of the suitcase. The magnitude of the normal force is equal to the magnitude of the weight of the suitcase minus the magnitude of the force of the pull. The magnitude of the normal force is equal to the sum of the magnitude of the force of the pull and the magnitude of the suitcase’s weight. The magnitude of the normal force is greater than the magnitude of the weight of the suitcase. Typesetting math: 100% Now assume that the man of weight is tired and decides to sit on his suitcase. Which statement about the magnitude of the normal force acting on the suitcase is true during the time that the man is sitting on the suitcase? Hint 1. Identify the correct free-body diagram. Which of the figures represents the free-body diagram while the man is sitting atop the suitcase? Here the vector labeled is a force that has the same magnitude as the man’s weight. ANSWER: wm n wm Typesetting math: 100% ANSWER: Correct Recognize that the normal force acting on an object is not always equal to the weight of that object. This is an important point to understand. Problem 6.5 A construction worker with a weight of 880 stands on a roof that is sloped at 18 . Part A What is the magnitude of the normal force of the roof on the worker? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct A B C D The magnitude of the normal force is equal to the magnitude of the suitcase’s weight. The magnitude of the normal force is equal to the magnitude of the suitcase’s weight minus the magnitude of the man’s weight. The magnitude of the normal force is equal to the sum of the magnitude of the man’s weight and the magnitude of the suitcase’s weight. The magnitude of the normal force is less than the magnitude of the suitcase’s weight. N  n = 840 N Typesetting math: 100% Problem 6.6 In each of the two free-body diagrams, the forces are acting on a 3.0 object. Part A For diagram , find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B For diagram the part A, find the value of the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: kg ax x ax = -0.67 m s2 ay, y Typesetting math: 100% Correct Part C For diagram , find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D For diagram the part C, find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: ay = 0 m s2 ax x ax = 0.67 m s2 ay y Typesetting math: 100% Correct Problem 6.7 In each of the two free-body diagrams, the forces are acting on a 3.0 object. Part A Find the value of , the component of the acceleration in diagram (a). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct ay = 0 m s2 kg ax x ax = 0.99 m s2 Typesetting math: 100% Part B Find the value of , the component of the acceleration in diagram (a). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C Find the value of , the component of the acceleration in diagram (b). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D Find the value of , the component of the acceleration in diagram (b). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct ay y ay = 0 m s2 ax x ax = -0.18 m s2 ay y ay = 0 m s2 Typesetting math: 100% Problem 6.10 A horizontal rope is tied to a 53.0 box on frictionless ice. What is the tension in the rope if: Part A The box is at rest? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part B The box moves at a steady = 4.80 ? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part C The box = 4.80 and = 4.60 ? Express your answer to three significant figures and include the appropriate units. ANSWER: kg T = 0 N vx m/s T = 0 N vx m/s ax m/s2 Typesetting math: 100% Correct Problem 6.14 It takes the elevator in a skyscraper 4.5 to reach its cruising speed of 11 . A 60 passenger gets aboard on the ground floor. Part A What is the passenger’s weight before the elevator starts moving? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the passenger’s weight while the elevator is speeding up? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the passenger’s weight after the elevator reaches its cruising speed? T = 244 N s m/s kg w = 590 N w = 730 N Typesetting math: 100% Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Block on an Incline A block lies on a plane raised an angle from the horizontal. Three forces act upon the block: , the force of gravity; , the normal force; and , the force of friction. The coefficient of friction is large enough to prevent the block from sliding . Part A Consider coordinate system a, with the x axis along the plane. Which forces lie along the axes? ANSWER: w = 590 N  F  w F n F  f Typesetting math: 100% Correct Part B Which forces lie along the axes of the coordinate system b, in which the y axis is vertical? ANSWER: Correct only only only and and and and and F  f F  n F  w F  f F  n F  f F  w F  n F w F  f F  n F w only only only and and and and and F  f F  n F  w F  f F  n F  f F  w F  n F w F  f F  n F w Typesetting math: 100% Usually the best advice is to choose coordinate system so that the acceleration of the system is directly along one of the coordinate axes. If the system isn’t accelerating, then you are better off choosing the coordinate system with the most vectors along the coordinate axes. But now you are going to ignore that advice. You will find the normal force, , using vertical coordinate system b. In these coordinates you will find the magnitude appearing in both the x and y equations, each multiplied by a trigonometric function. Part C Because the block is not moving, the sum of the y components of the forces acting on the block must be zero. Find an expression for the sum of the y components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables , , , and . Hint 1. Find the y component of Write an expression for , the y component of the force , using coordinate system b. Express your answer in terms of and . Hint 1. Some geometry help – a useful angle The smaller angle between and the y-axis is also , as shown in the figure. ANSWER: F  n Fn Fn Ff Fw  F n Fny F  n Fn  F  n  Typesetting math: 100% Hint 2. Find the y component of Write an expression for , the y component of the force , using coordinate system b. Express your answer in terms of and . Hint 1. Some geometry help – a useful angle The smaller angle between and the x-axis is also , as shown in the figure. ANSWER: ANSWER: Fny = Fncos() F f Ffy F f Ff  F  f  Ffy = Ffsin() Fy = 0 = Fncos() + Ffsin() − Fw Typesetting math: 100% Correct Part D Because the block is not moving, the sum of the x components of the forces acting on the block must be zero. Find an expression for the sum of the x components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables , , , and . Hint 1. Find the x component of Write an expression for , the x component of the force , using coordinate system b. Express your answer in terms of and . ANSWER: ANSWER: Correct Part E To find the magnitude of the normal force, you must express in terms of since is an unknown. Using the equations you found in the two previous parts, find an expression for involving and but not . Hint 1. How to approach the problem From your answers to the previous two parts you should have two force equations ( and ). Combine these equations to eliminate . The key is to multiply the Fn Ff Fw  F n Fnx F  n Fn  Fnx = −Fnsin() Fx = 0 = −Fnsin() + Ffcos() Fn Fw Ff Fn Fw  Ff Typesetting math: 100% Fy = 0 Fx = 0 Ff equation for the y components by and the equation for the x components by , then add or subtract the two equations to eliminate the term . An alternative motivation for the algebra is to eliminate the trig functions in front of by using the trig identity . At the very least this would result in an equation that is simple to solve for . ANSWER: Correct Congratulations on working this through. Now realize that in coordinate system a, which is aligned with the plane, the y-coordinate equation is , which leads immediately to the result obtained here for . CONCLUSION: A thoughtful examination of which coordinate system to choose can save a lot of algebra. Contact Forces Introduced Learning Goal: To introduce contact forces (normal and friction forces) and to understand that, except for friction forces under certain circumstances, these forces must be determined from: net Force = ma. Two solid objects cannot occupy the same space at the same time. Indeed, when the objects touch, they exert repulsive normal forces on each other, as well as frictional forces that resist their slipping relative to each other. These contact forces arise from a complex interplay between the electrostatic forces between the electrons and ions in the objects and the laws of quantum mechanics. As two surfaces are pushed together these forces increase exponentially over an atomic distance scale, easily becoming strong enough to distort the bulk material in the objects if they approach too close. In everyday experience, contact forces are limited by the deformation or acceleration of the objects, rather than by the fundamental interatomic forces. Hence, we can conclude the following: The magnitude of contact forces is determined by , that is, by the other forces on, and acceleration of, the contacting bodies. The only exception is that the frictional forces cannot exceed (although they can be smaller than this or even zero). Normal and friction forces Two types of contact forces operate in typical mechanics problems, the normal and frictional forces, usually designated by and (or , or something similar) respectively. These are the components of the overall contact force: perpendicular to and parallel to the plane of contact. Kinetic friction when surfaces slide cos  sin  Ff cos() sin() Fn sin2() + cos2 () = 1 Fn Fn = Fwcos() Fy = Fn − FW cos() = 0 Fn F = ma μn n f Ffric n f Typesetting math: 100% When one surface is sliding past the other, experiments show three things about the friction force (denoted ): The frictional force opposes the relative motion at the 1. point of contact, 2. is proportional to the normal force, and 3. the ratio of the magnitude of the frictional force to that of the normal force is fairly constant over a wide range of speeds. The constant of proportionality is called the coefficient of kinetic friction, often designated . As long as the sliding continues, the frictional force is then (valid when the surfaces slide by each other). Static friction when surfaces don’t slide When there is no relative motion of the surfaces, the frictional force can assume any value from zero up to a maximum , where is the coefficient of static friction. Invariably, is larger than , in agreement with the observation that when a force is large enough that something breaks loose and starts to slide, it often accelerates. The frictional force for surfaces with no relative motion is therefore (valid when the contacting surfaces have no relative motion). The actual magnitude and direction of the static friction force are such that it (together with other forces on the object) causes the object to remain motionless with respect to the contacting surface as long as the static friction force required does not exceed . The equation is valid only when the surfaces are on the verge of sliding. Part A When two objects slide by one another, which of the following statements about the force of friction between them, is true? ANSWER: Correct Part B fk fk μk fk = μkn μsn μs μs μk fs ! μsn μsn fs = μsn The frictional force is always equal to . The frictional force is always less than . The frictional force is determined by other forces on the objects so it can be either equal to or less than . μkn μkn μkn Typesetting math: 100% When two objects are in contact with no relative motion, which of the following statements about the frictional force between them, is true? ANSWER: Correct For static friction, the actual magnitude and direction of the friction force are such that it, together with any other forces present, will cause the object to have the observed acceleration. The magnitude of the force cannot exceed . If the magnitude of static friction needed to keep acceleration equal to zero exceeds , then the object will slide subject to the resistance of kinetic friction. Do not automatically assume that unless you are considering a situation in which the magnitude of the static friction force is as large as possible (i.e., when determining at what point an object will just begin to slip). Whether the actual magnitude of the friction force is 0, less than , or equal to depends on the magnitude of the other forces (if any) as well as the acceleration of the object through . Part C When a board with a box on it is slowly tilted to larger and larger angle, common experience shows that the box will at some point “break loose” and start to accelerate down the board. The box begins to slide once the component of gravity acting parallel to the board just begins to exceeds the maximum force of static friction. Which of the following is the most general explanation for why the box accelerates down the board? ANSWER: The frictional force is always equal to . The frictional force is always less than . The frictional force is determined by other forces on the objects so it can be either equal to or less than . μsn μsn μsn μsn μsn fs = μsn μsn μsn F = ma Fg The force of kinetic friction is smaller than that of maximum static friction, but remains the same. Once the box is moving, is smaller than the force of maximum static friction but larger than the force of kinetic friction. Once the box is moving, is larger than the force of maximum static friction. When the box is stationary, equals the force of static friction, but once the box starts moving, the sliding reduces the normal force, which in turn reduces the friction. Fg Fg Fg Fg Typesetting math: 100% Correct At the point when the box finally does “break loose,” you know that the component of the box’s weight that is parallel to the board just exceeds (i.e., this component of gravitational force on the box has just reached a magnitude such that the force of static friction, which has a maximum value of , can no longer oppose it.) For the box to then accelerate, there must be a net force on the box along the board. Thus, the component of the box’s weight parallel to the board must be greater than the force of kinetic friction. Therefore the force of kinetic friction must be less than the force of static friction which implies , as expected. Part D Consider a problem in which a car of mass is on a road tilted at an angle . The normal force Select the best answer. ANSWER: Correct The key point is that contact forces must be determined from Newton’s equation. In the problem described above, there is not enough information given to determine the normal force (e.g., the acceleration is unknown). Each of the answer options is valid under some conditions ( , the car is sliding down an icy incline, or the car is going around a banked turn), but in fact none is likely to be correct if there are other forces on the car or if the car is accelerating. Do not memorize values for the normal force valid in different problems–you must determine from . Problem 6.17 Bonnie and Clyde are sliding a 323 bank safe across the floor to their getaway car. The safe slides with a constant speed if Clyde pushes from behind with 375 of force while Bonnie pulls forward on a rope with 335 of force. μsn μsn μkn μsn μk < μs M  is found using n = Mg n = Mg cos() n = Mg cos() F  = Ma  = 0 n F = ma kg N N Typesetting math: 100% Part A What is the safe's coefficient of kinetic friction on the bank floor? ANSWER: Correct Problem 6.19 A crate is placed on a horizontal conveyor belt. The materials are such that and . Part A Draw a free-body diagram showing all the forces on the crate if the conveyer belt runs at constant speed. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: 0.224 10 kg μs = 0.5 μk = 0.3 Typesetting math: 100% Correct Part B Draw a free-body diagram showing all the forces on the crate if the conveyer belt is speeding up. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: Typesetting math: 100% Correct Part C What is the maximum acceleration the belt can have without the crate slipping? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct amax = 4.9 m s2 Typesetting math: 100% Problem 6.28 A 1100 steel beam is supported by two ropes. Part A What is the tension in rope 1? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the tension in rope 2? Express your answer to two significant figures and include the appropriate units. ANSWER: kg T1 = 7000 N Typesetting math: 100% Correct Problem 6.35 The position of a 1.4 mass is given by , where is in seconds. Part A What is the net horizontal force on the mass at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the net horizontal force on the mass at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 6.39 T2 = 4800 N kg x = (2t3 − 3t2 )m t t = 0 s F = -8.4 N t = 1 s F = 8.4 N Typesetting math: 100% A rifle with a barrel length of 61 fires a 8 bullet with a horizontal speed of 400 . The bullet strikes a block of wood and penetrates to a depth of 11 . Part A What resistive force (assumed to be constant) does the wood exert on the bullet? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B How long does it take the bullet to come to rest after entering the wood? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 6.45 You and your friend Peter are putting new shingles on a roof pitched at 21 . You're sitting on the very top of the roof when Peter, who is at the edge of the roof directly below you, 5.0 away, asks you for the box of nails. Rather than carry the 2.0 box of nails down to Peter, you decide to give the box a push and have it slide down to him. Part A If the coefficient of kinetic friction between the box and the roof is 0.55, with what speed should you push the box to have it gently come to rest right at the edge of the roof? Express your answer to two significant figures and include the appropriate units. cm g m/s cm fk = 5800 N = 5.5×10−4 t s  m kg Typesetting math: 100% ANSWER: Correct Problem 6.54 The 2.0 wood box in the figure slides down a vertical wood wall while you push on it at a 45 angle. Part A What magnitude of force should you apply to cause the box to slide down at a constant speed? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct v = 3.9 ms kg  F = 23 N Typesetting math: 100% Score Summary: Your score on this assignment is 98.8%. You received 114.57 out of a possible total of 116 points. Typesetting math: 100%

Assignment 5 Due: 11:59pm on Wednesday, March 5, 2014 You … Read More...
An approximation for the boundary-layer shape in the formula where U is stream velocity far from the wall and s is the boundary layer thickness, if the fluid is helium at 20c I atm, and if u=10.8 m/s and s =3 mm, use the formula to (a) estimate the wall shear stress in pa ; and (b) find the position in boundary layer where is one half

An approximation for the boundary-layer shape in the formula where U is stream velocity far from the wall and s is the boundary layer thickness, if the fluid is helium at 20c I atm, and if u=10.8 m/s and s =3 mm, use the formula to (a) estimate the wall shear stress in pa ; and (b) find the position in boundary layer where is one half

The plate in figure is uniform and weigh 300 Ib. It is supported in a horizontal position by a hinge in the middle of CD and by a cable at point B. a) Draw a complete free body diagram for the plate . b) Determine the tension in the cable .

The plate in figure is uniform and weigh 300 Ib. It is supported in a horizontal position by a hinge in the middle of CD and by a cable at point B. a) Draw a complete free body diagram for the plate . b) Determine the tension in the cable .

 
STUDENT GRADER Total Score I am submitting my own work, and I understand penalties will be assessed if I submit work for credit that is not my own. Print Name ID Number Sign Name Date # Points Score 1 4 2 8 3 6 4 12 5 4 6 10 7 8 8 6 9 6 Weeks late Adjusted Score Estimated Work Hours 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Overall Weight Adjusted Score: Deduct 20% from score for each week late Problem 1. Sketch circuits for the following logic equations. Y <= (A and B and C) or not ((A and not B and C and not D) or not (B or D)); X <= (A xor (B and C) xor not D) or (not (B xor C) and not (C or D)) Problem 2. Sketch circuits and write VHDL assignment statements for the following equations. F = m(1, 2, 6) F = M(0, 7) Problem 3. Write logic assignment statements for the following circuit. Problem 4: Sketch circuits and write VHDL assignment statements for the truth tables below. Problem 5: Sketch POS circuits for the 2XOR and 2XNOR functions. Problem 6: Sketch the circuit described by the netlist shown, and complete the timing diagram for the stimulus shown to document the circuit’s response to the example stimulus. Use a 100ns vertical grid in your timing diagram, and show all inputs and outputs. Problem 7: Create a truth table that corresponds to the simulation shown below. Show all input and output values in the truth table, and sketch a logic circuit that could have been used to create the waveform. Problem 8. The Seattle Mariners haven’t had a stolen base in 6 months, and the manager decided it was because the other teams were reading his signals to the base runners. He came up with a new set of signals (pulling on his EAR, lifting one LEG, patting the top of his HEAD, and BOWing) to indicate when runners should attempt to steal a base. A runner should STEAL a base if and only if the manager pulls his EAR and BOWs while patting his HEAD, or if he lifts his LEG and pats his HEAD without BOWing, or anytime he pulls his EAR without lifting his LEG. Sketch a minimal circuit that could be used to indicate when a runner should steal a base. Problem 9. A room has four doors and four light switches (one by each door). Sketch a circuit that allows the four switches to control the light – each switch should be able to turn the light on if it is currently off, and off if it is currently on. Note that it will not be possible to associate a given switch position with “light on” or “light off” – simply moving any switch should modify the light’s status.

STUDENT GRADER Total Score I am submitting my own work, and I understand penalties will be assessed if I submit work for credit that is not my own. Print Name ID Number Sign Name Date # Points Score 1 4 2 8 3 6 4 12 5 4 6 10 7 8 8 6 9 6 Weeks late Adjusted Score Estimated Work Hours 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Overall Weight Adjusted Score: Deduct 20% from score for each week late Problem 1. Sketch circuits for the following logic equations. Y <= (A and B and C) or not ((A and not B and C and not D) or not (B or D)); X <= (A xor (B and C) xor not D) or (not (B xor C) and not (C or D)) Problem 2. Sketch circuits and write VHDL assignment statements for the following equations. F = m(1, 2, 6) F = M(0, 7) Problem 3. Write logic assignment statements for the following circuit. Problem 4: Sketch circuits and write VHDL assignment statements for the truth tables below. Problem 5: Sketch POS circuits for the 2XOR and 2XNOR functions. Problem 6: Sketch the circuit described by the netlist shown, and complete the timing diagram for the stimulus shown to document the circuit’s response to the example stimulus. Use a 100ns vertical grid in your timing diagram, and show all inputs and outputs. Problem 7: Create a truth table that corresponds to the simulation shown below. Show all input and output values in the truth table, and sketch a logic circuit that could have been used to create the waveform. Problem 8. The Seattle Mariners haven’t had a stolen base in 6 months, and the manager decided it was because the other teams were reading his signals to the base runners. He came up with a new set of signals (pulling on his EAR, lifting one LEG, patting the top of his HEAD, and BOWing) to indicate when runners should attempt to steal a base. A runner should STEAL a base if and only if the manager pulls his EAR and BOWs while patting his HEAD, or if he lifts his LEG and pats his HEAD without BOWing, or anytime he pulls his EAR without lifting his LEG. Sketch a minimal circuit that could be used to indicate when a runner should steal a base. Problem 9. A room has four doors and four light switches (one by each door). Sketch a circuit that allows the four switches to control the light – each switch should be able to turn the light on if it is currently off, and off if it is currently on. Note that it will not be possible to associate a given switch position with “light on” or “light off” – simply moving any switch should modify the light’s status.

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Extra Credit Due: 11:59pm on Thursday, May 15, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy A Man Running to Catch a Bus A man is running at speed (much less than the speed of light) to catch a bus already at a stop. At , when he is a distance from the door to the bus, the bus starts moving with the positive acceleration . Use a coordinate system with at the door of the stopped bus. Part A What is , the position of the man as a function of time? Answer symbolically in terms of the variables , , and . Hint 1. Which equation should you use for the man’s speed? Because the man’s speed is constant, you may use . ANSWER: c t = 0 b a x = 0 xman(t) b c t x(t) = x(0) + vt xman(t) = −b + ct Correct Part B What is , the position of the bus as a function of time? Answer symbolically in terms of and . Hint 1. Which equation should you use for the bus’s acceleration? Because the bus has constant acceleration, you may use . Recall that . ANSWER: Correct Part C What condition is necessary for the man to catch the bus? Assume he catches it at time . Hint 1. How to approach this problem If the man is to catch the bus, then at some moment in time , the man must arrive at the position of the door of the bus. How would you express this condition mathematically? ANSWER: xbus(t) a t x(t) = x(0) + v(0)t + (1/2)at2 vbus(0) = 0 xbus = 1 a 2 t2 tcatch tcatch Typesetting math: 15% Correct Part D Inserting the formulas you found for and into the condition , you obtain the following: , or . Intuitively, the man will not catch the bus unless he is running fast enough. In mathematical terms, there is a constraint on the man’s speed so that the equation above gives a solution for that is a real positive number. Find , the minimum value of for which the man will catch the bus. Express the minimum value for the man’s speed in terms of and . Hint 1. Consider the discriminant Use the quadratic equation to solve: . What is the discriminant (the part under the radical) of the solution for ? xman(tcatch) > xbus(tcatch) xman(tcatch) = xbus(tcatch) xman(tcatch) < xbus(tcatch) c = a  tcatch xman(t) xbus(t) xman(tcatch) = xbus(tcatch) −b+ct = a catch 1 2 t2 catch 1 a −c +b = 0 2 t2 catch tcatch c tcatch cmin c a b 1 a − c + b = 0 2 t2 catch tcatch tcatch Typesetting math: 15% Hint 1. The quadratic formula Recall: If then ANSWER: Hint 2. What is the constraint? To get a real value for , the discriminant must be greater then or equal to zero. This condition yields a constraint that exceed . ANSWER: Correct Part E Assume that the man misses getting aboard when he first meets up with the bus. Does he get a second chance if he continues to run at the constant speed ? Hint 1. What is the general quadratic equation? The general quadratic equation is , where , \texttip{B}{B}, and \texttip{C}{C} are constants. Depending on the value of the discriminant, \Delta = c^2-2ab, the equation may have Ax2 + Bx + C = 0 x = −B±B2−4AC 2A  = cc − 2ab tcatch c cmin cmin = (2ab) −−−−  c > cmin Ax2 + Bx + C = 0 A Typesetting math: 15% two real valued solutions 1. if \Delta > 0, 2. one real valued solution if \Delta = 0, or 3. two complex valued solutions if \Delta < 0. In this case, every real valued solution corresponds to a time at which the man is at the same position as the door of the bus. ANSWER: Correct Adding and Subtracting Vectors Conceptual Question Six vectors (A to F) have the magnitudes and directions indicated in the figure. Part A No; there is no chance he is going to get aboard. Yes; he will get a second chance Typesetting math: 15% Which two vectors, when added, will have the largest (positive) x component? Hint 1. Largest x component The two vectors with the largest x components will, when combined, give the resultant with the largest x component. Keep in mind that positive x components are larger than negative x components. ANSWER: Correct Part B Which two vectors, when added, will have the largest (positive) y component? Hint 1. Largest y component The two vectors with the largest y components will, when combined, give the resultant with the largest y component. Keep in mind that positive y components are larger than negative y components. ANSWER: C and E E and F A and F C and D B and D Typesetting math: 15% Correct Part C Which two vectors, when subtracted (i.e., when one vector is subtracted from the other), will have the largest magnitude? Hint 1. Subtracting vectors To subtract two vectors, add a vector with the same magnitude but opposite direction of one of the vectors to the other vector. ANSWER: Correct Tactics Box 3.1 Determining the Components of a Vector Learning Goal: C and D A and F E and F A and B E and D A and F A and E D and B C and D E and F Typesetting math: 15% To practice Tactics Box 3.1 Determining the Components of a Vector. When a vector \texttip{\vec{A}}{A_vec} is decomposed into component vectors \texttip{\vec{A}_{\mit x}}{A_vec_x} and \texttip{\vec{A}_{\mit y}}{A_vec_y} parallel to the coordinate axes, we can describe each component vector with a single number (a scalar) called the component. This tactics box describes how to determine the x component and y component of vector \texttip{\vec{A}}{A_vec}, denoted \texttip{A_{\mit x}}{A_x} and \texttip{A_{\mit y}}{A_y}. TACTICS BOX 3.1 Determining the components of a vector The absolute value |A_x| of the x component \texttip{A_{\mit x}}{A_x} is the magnitude of the component vector \texttip{\vec{A}_{\1. mit x}}{A_vec_x}. The sign of \texttip{A_{\mit x}}{A_x} is positive if \texttip{\vec{A}_{\mit x}}{A_vec_x} points in the positive x direction; it is negative if \texttip{\vec{A}_{\mit x}}{A_vec_x} points in the negative x direction. 2. 3. The y component \texttip{A_{\mit y}}{A_y} is determined similarly. Part A What is the magnitude of the component vector \texttip{\vec{A}_{\mit x}}{A_vec_x} shown in the figure? Express your answer in meters to one significant figure. ANSWER: Correct |A_x| = 5 \rm m Typesetting math: 15% Part B What is the sign of the y component \texttip{A_{\mit y}}{A_y} of vector \texttip{\vec{A}}{A_vec} shown in the figure? ANSWER: Correct Part C Now, combine the information given in the tactics box above to find the x and y components, \texttip{B_{\mit x}}{B_x} and \texttip{B_{\mit y}}{B_y}, of vector \texttip{\vec{B}}{B_vec} shown in the figure. Express your answers, separated by a comma, in meters to one significant figure. positive negative Typesetting math: 15% ANSWER: Correct Conceptual Problem about Projectile Motion Learning Goal: To understand projectile motion by considering horizontal constant velocity motion and vertical constant acceleration motion independently. Projectile motion refers to the motion of unpowered objects (called projectiles) such as balls or stones moving near the surface of the earth under the influence of the earth's gravity alone. In this analysis we assume that air resistance can be neglected. An object undergoing projectile motion near the surface of the earth obeys the following rules: An object undergoing projectile motion travels horizontally at a constant rate. That is, the x component of its velocity, \texttip{v_{\mit x}}{1. v_x}, is constant. An object undergoing projectile motion moves vertically with a constant downward acceleration whose magnitude, denoted by \texttip{g}{g}, is equal to 9.80 \rm{m/s^2} near the surface of the earth. Hence, the y component of its velocity, \texttip{v_{\mit y}}{v_y}, changes continuously. 2. An object undergoing projectile motion will undergo the horizontal and vertical motions described above from the instant it is launched until the instant it strikes the ground again. Even though the horizontal and vertical motions can be treated independently, they are related by the fact that they occur for exactly the same amount of time, namely the time \texttip{t}{t} the projectile is in the air. 3. The figure shows the trajectory (i.e., the path) of a ball undergoing projectile motion over level ground. The time t_0 = 0\;\rm{s} corresponds to the moment just after the ball is launched from position x_0 = 0\;\rm{m} and y_0 = 0\;\rm{m}. Its launch velocity, also called the initial velocity, is \texttip{\vec{v}_{\rm 0}}{v_vec_0}. Two other points along the trajectory are indicated in the figure. One is the moment the ball reaches the peak of its trajectory, at time \texttip{t_{\rm 1}}{t_1} with velocity \texttip{\vec{v}_{\rm 1}}{v_1_vec}. Its position at this moment is denoted by (x_1, y_1) or (x_1, y_{\max}) since it is at its maximum \texttip{B_{\mit x}}{B_x}, \texttip{B_{\mit y}}{B_y} = -2,-5 \rm m, \rm m Typesetting math: 15% The other point, at time \texttip{t_{\rm 2}}{t_2} with velocity \texttip{\vec{v}_{\rm 2}}{v_2_vec}, corresponds to the moment just before the ball strikes the ground on the way back down. At this time its position is (x_2, y_2), also known as (x_{\max}, y_2) since it is at its maximum horizontal range. Projectile motion is symmetric about the peak, provided the object lands at the same vertical height from which is was launched, as is the case here. Hence y_2 = y_0 = 0\;\rm{m}. Part A How do the speeds \texttip{v_{\rm 0}}{v_0}, \texttip{v_{\rm 1}}{v_1}, and \texttip{v_{\rm 2}}{v_2} (at times \texttip{t_{\rm 0}}{t_0}, \texttip{t_{\rm 1}}{t_1}, and \texttip{t_{\rm 2}}{t_2}) compare? ANSWER: Correct Here \texttip{v_{\rm 0}}{v_0} equals \texttip{v_{\rm 2}}{v_2} by symmetry and both exceed \texttip{v_{\rm 1}}{v_1}. This is because \texttip{v_{\rm 0}}{v_0} and \texttip{v_{\rm 2}}{v_2} include vertical speed as well as the constant horizontal speed. Consider a diagram of the ball at time \texttip{t_{\rm 0}}{t_0}. Recall that \texttip{t_{\rm 0}}{t_0} refers to the instant just after the ball has been launched, so it is still at ground level (x_0 = y_0= 0\;\rm{m}). However, it is already moving with initial velocity \texttip{\vec{v}_{\rm 0}}{v_0_vec}, whose magnitude is v_0 = 30.0\;{\rm m/s} and direction is \theta = 60.0\;{\rm degrees} counterclockwise from the positive x direction. \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 1}}{v_1} = \texttip{v_{\rm 2}}{v_2} > 0 \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} = 0 \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} > 0 \texttip{v_{\rm 0}}{v_0} > \texttip{v_{\rm 1}}{v_1} > \texttip{v_{\rm 2}}{v_2} > 0 \texttip{v_{\rm 0}}{v_0} > \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} = 0 Typesetting math: 15% Part B What are the values of the intial velocity vector components \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} (both in \rm{m/s}) as well as the acceleration vector components \texttip{a_{0,x}}{a_0, x} and \texttip{a_{0,y}}{a_0, y} (both in \rm{m/s^2})? Here the subscript 0 means “at time \texttip{t_{\rm 0}}{t_0}.” Hint 1. Determining components of a vector that is aligned with an axis If a vector points along a single axis direction, such as in the positive x direction, its x component will be its full magnitude, whereas its y component will be zero since the vector is perpendicular to the y direction. If the vector points in the negative x direction, its x component will be the negative of its full magnitude. Hint 2. Calculating the components of the initial velocity Notice that the vector \texttip{\vec{v}_{\rm 0}}{v_0_vec} points up and to the right. Since “up” is the positive y axis direction and “to the right” is the positive x axis direction, \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} will both be positive. As shown in the figure, \texttip{v_{0,x}}{v_0, x}, \texttip{v_{0,y}}{v_0, y}, and \texttip{v_{\rm 0}}{v_0} are three sides of a right triangle, one angle of which is \texttip{\theta }{theta}. Thus \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} can be found using the definition of the sine and cosine functions given below. Recall that v_0 = 30.0\;\rm{m/s} and \theta = 60.0\;\rm{degrees} and note that \large{\sin(\theta) = \frac{\rm{length\;of\;opposite\;side}}{\rm{length\;of\;hypotenuse}}} \large{= \frac{v_{0, y}}{v_0}}, \large{\cos(\theta) = \frac{\rm{length\;of\;adjacent\;side}}{\rm{length\;of\;hypotenuse}}} \large{= \frac{v_{0, x}}{v_0}.} What are the values of \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y}? Enter your answers numerically in meters per second separated by a comma. ANSWER: ANSWER: 15.0,26.0 \rm{m/s} Typesetting math: 15% Correct Also notice that at time \texttip{t_{\rm 2}}{t_2}, just before the ball lands, its velocity components are v_{2, x} = 15\;\rm{m/s} (the same as always) and v_{2, y} = – 26.0\;\rm{m/s} (the same size but opposite sign from \texttip{v_{0,y}}{v_0, y} by symmetry). The acceleration at time \texttip{t_{\rm 2}}{t_2} will have components (0, -9.80 \rm{m/s^2}), exactly the same as at \texttip{t_{\rm 0}}{t_0}, as required by Rule 2. The peak of the trajectory occurs at time \texttip{t_{\rm 1}}{t_1}. This is the point where the ball reaches its maximum height \texttip{y_{\rm max}}{y_max}. At the peak the ball switches from moving up to moving down, even as it continues to travel horizontally at a constant rate. Part C What are the values of the velocity vector components \texttip{v_{1,x}}{v_1, x} and \texttip{v_{1,y}}{v_1, y} (both in \rm{m/s}) as well as the acceleration vector components \texttip{a_{1,x}}{a_1, x} and \texttip{a_{1,y}}{a_1, y} (both in \rm{m/s^2})? Here the subscript 1 means that these are all at time \texttip{t_{\rm 1}}{t_1}. ANSWER: 30.0, 0, 0, 0 0, 30.0, 0, 0 15.0, 26.0, 0, 0 30.0, 0, 0, -9.80 0, 30.0, 0, -9.80 15.0, 26.0, 0, -9.80 15.0, 26.0, 0, +9.80 Typesetting math: 15% Correct At the peak of its trajectory the ball continues traveling horizontally at a constant rate. However, at this moment it stops moving up and is about to move back down. This constitutes a downward-directed change in velocity, so the ball is accelerating downward even at the peak. The flight time refers to the total amount of time the ball is in the air, from just after it is launched (\texttip{t_{\rm 0}}{t_0}) until just before it lands (\texttip{t_{\rm 2}}{t_2}). Hence the flight time can be calculated as t_2 – t_0, or just \texttip{t_{\rm 2}}{t_2} in this particular situation since t_0 = 0. Because the ball lands at the same height from which it was launched, by symmetry it spends half its flight time traveling up to the peak and the other half traveling back down. The flight time is determined by the initial vertical component of the velocity and by the acceleration. The flight time does not depend on whether the object is moving horizontally while it is in the air. Part D If a second ball were dropped from rest from height \texttip{y_{\rm max}}{y_max}, how long would it take to reach the ground? Ignore air resistance. Check all that apply. Hint 1. Kicking a ball of cliff; a related problem Consider two balls, one of which is dropped from rest off the edge of a cliff at the same moment that the other is kicked horizontally off the edge of the cliff. Which ball reaches the level ground at the base of the cliff first? Ignore air resistance. Hint 1. Comparing position, velocity, and acceleration of the two balls Both balls start at the same height and have the same initial y velocity (v_{0,y} = 0) as well as the same acceleration (\vec a = g downward). They differ only in their x velocity (one is 0, 0, 0, 0 0, 0, 0, -9.80 15.0, 0, 0, 0 15.0, 0, 0, -9.80 0, 26.0, 0, 0 0, 26.0, 0, -9.80 15.0, 26.0, 0, 0 15.0, 26.0, 0, -9.80 Typesetting math: 15% zero, the other nonzero). This difference will affect their x motion but not their y motion. ANSWER: ANSWER: Correct In projectile motion over level ground, it takes an object just as long to rise from the ground to the peak as it takes for it to fall from the peak back to the ground. The range \texttip{R}{R} of the ball refers to how far it moves horizontally, from just after it is launched until just before it lands. Range is defined as x_2 – x_0, or just \texttip{x_{\rm 2}}{x_2} in this particular situation since x_0 = 0. Range can be calculated as the product of the flight time \texttip{t_{\rm 2}}{t_2} and the x component of the velocity \texttip{v_{\mit x}}{v_x} (which is the same at all times, so v_x = v_{0,x}). The value of \texttip{v_{\mit x}}{v_x} can be found from the launch speed \texttip{v_{\rm 0}}{v_0} and the launch angle \texttip{\theta }{theta} using trigonometric functions, as was done in Part B. The flight time is related to the initial y component of the velocity, which may also be found from \texttip{v_{\rm 0}}{v_0} and \texttip{\theta }{theta} using trig functions. The following equations may be useful in solving projectile motion problems, but these equations apply only to a projectile launched over level ground from position (x_0 = y_0 = 0) at time t_0 = 0 with initial speed \texttip{v_{\rm 0}}{v_0} and launch angle \texttip{\theta }{theta} measured from the horizontal. As was the case above, \texttip{t_{\rm 2}}{t_2} refers to the flight time and \texttip{R}{R} refers to the range of the projectile. flight time: \large{t_2 = \frac{2 v_{0, y}}{g} = \frac{2 v_0 \sin(\theta)}{g}} range: \large{R = v_x t_2 = \frac{v_0^2 \sin(2\theta)}{g}} The ball that falls straight down strikes the ground first. The ball that was kicked so it moves horizontally as it falls strikes the ground first. Both balls strike the ground at the same time. \texttip{t_{\rm 0}}{t_0} t_1 – t_0 \texttip{t_{\rm 2}}{t_2} t_2 – t_1 \large{\frac{t_2 – t_0}{2}} Typesetting math: 15% In general, a high launch angle yields a long flight time but a small horizontal speed and hence little range. A low launch angle gives a larger horizontal speed, but less flight time in which to accumulate range. The launch angle that achieves the maximum range for projectile motion over level ground is 45 degrees. Part E Which of the following changes would increase the range of the ball shown in the original figure? Check all that apply. ANSWER: Correct A solid understanding of the concepts of projectile motion will take you far, including giving you additional insight into the solution of projectile motion problems numerically. Even when the object does not land at the same height from which is was launched, the rules given in the introduction will still be useful. Recall that air resistance is assumed to be negligible here, so this projectile motion analysis may not be the best choice for describing things like frisbees or feathers, whose motion is strongly influenced by air. The value of the gravitational free-fall acceleration \texttip{g}{g} is also assumed to be constant, which may not be appropriate for objects that move vertically through distances of hundreds of kilometers, like rockets or missiles. However, for problems that involve relatively dense projectiles moving close to the surface of the earth, these assumptions are reasonable. A World-Class Sprinter World-class sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has magnitude 15 \;{\rm m}/{\rm s}^{2}. Part A How much horizontal force \texttip{F}{F} must a sprinter of mass 64{\rm kg} exert on the starting blocks to produce this acceleration? Express your answer in newtons using two significant figures. Increase \texttip{v_{\rm 0}}{v_0} above 30 \rm{m/s}. Reduce \texttip{v_{\rm 0}}{v_0} below 30 \rm{m/s}. Reduce \texttip{\theta }{theta} from 60 \rm{degrees} to 45 \rm{degrees}. Reduce \texttip{\theta }{theta} from 60 \rm{degrees} to less than 30 \rm{degrees}. Increase \texttip{\theta }{theta} from 60 \rm{degrees} up toward 90 \rm{degrees}. Typesetting math: 15% Hint 1. Newton’s 2nd law of motion According to Newton’s 2nd law of motion, if a net external force \texttip{F_{\rm net}}{F_net} acts on a body, the body accelerates, and the net force is equal to the mass \texttip{m}{m} of the body times the acceleration \texttip{a}{a} of the body: F_{\rm net} = ma. ANSWER: Co

Extra Credit Due: 11:59pm on Thursday, May 15, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy A Man Running to Catch a Bus A man is running at speed (much less than the speed of light) to catch a bus already at a stop. At , when he is a distance from the door to the bus, the bus starts moving with the positive acceleration . Use a coordinate system with at the door of the stopped bus. Part A What is , the position of the man as a function of time? Answer symbolically in terms of the variables , , and . Hint 1. Which equation should you use for the man’s speed? Because the man’s speed is constant, you may use . ANSWER: c t = 0 b a x = 0 xman(t) b c t x(t) = x(0) + vt xman(t) = −b + ct Correct Part B What is , the position of the bus as a function of time? Answer symbolically in terms of and . Hint 1. Which equation should you use for the bus’s acceleration? Because the bus has constant acceleration, you may use . Recall that . ANSWER: Correct Part C What condition is necessary for the man to catch the bus? Assume he catches it at time . Hint 1. How to approach this problem If the man is to catch the bus, then at some moment in time , the man must arrive at the position of the door of the bus. How would you express this condition mathematically? ANSWER: xbus(t) a t x(t) = x(0) + v(0)t + (1/2)at2 vbus(0) = 0 xbus = 1 a 2 t2 tcatch tcatch Typesetting math: 15% Correct Part D Inserting the formulas you found for and into the condition , you obtain the following: , or . Intuitively, the man will not catch the bus unless he is running fast enough. In mathematical terms, there is a constraint on the man’s speed so that the equation above gives a solution for that is a real positive number. Find , the minimum value of for which the man will catch the bus. Express the minimum value for the man’s speed in terms of and . Hint 1. Consider the discriminant Use the quadratic equation to solve: . What is the discriminant (the part under the radical) of the solution for ? xman(tcatch) > xbus(tcatch) xman(tcatch) = xbus(tcatch) xman(tcatch) < xbus(tcatch) c = a  tcatch xman(t) xbus(t) xman(tcatch) = xbus(tcatch) −b+ct = a catch 1 2 t2 catch 1 a −c +b = 0 2 t2 catch tcatch c tcatch cmin c a b 1 a − c + b = 0 2 t2 catch tcatch tcatch Typesetting math: 15% Hint 1. The quadratic formula Recall: If then ANSWER: Hint 2. What is the constraint? To get a real value for , the discriminant must be greater then or equal to zero. This condition yields a constraint that exceed . ANSWER: Correct Part E Assume that the man misses getting aboard when he first meets up with the bus. Does he get a second chance if he continues to run at the constant speed ? Hint 1. What is the general quadratic equation? The general quadratic equation is , where , \texttip{B}{B}, and \texttip{C}{C} are constants. Depending on the value of the discriminant, \Delta = c^2-2ab, the equation may have Ax2 + Bx + C = 0 x = −B±B2−4AC 2A  = cc − 2ab tcatch c cmin cmin = (2ab) −−−−  c > cmin Ax2 + Bx + C = 0 A Typesetting math: 15% two real valued solutions 1. if \Delta > 0, 2. one real valued solution if \Delta = 0, or 3. two complex valued solutions if \Delta < 0. In this case, every real valued solution corresponds to a time at which the man is at the same position as the door of the bus. ANSWER: Correct Adding and Subtracting Vectors Conceptual Question Six vectors (A to F) have the magnitudes and directions indicated in the figure. Part A No; there is no chance he is going to get aboard. Yes; he will get a second chance Typesetting math: 15% Which two vectors, when added, will have the largest (positive) x component? Hint 1. Largest x component The two vectors with the largest x components will, when combined, give the resultant with the largest x component. Keep in mind that positive x components are larger than negative x components. ANSWER: Correct Part B Which two vectors, when added, will have the largest (positive) y component? Hint 1. Largest y component The two vectors with the largest y components will, when combined, give the resultant with the largest y component. Keep in mind that positive y components are larger than negative y components. ANSWER: C and E E and F A and F C and D B and D Typesetting math: 15% Correct Part C Which two vectors, when subtracted (i.e., when one vector is subtracted from the other), will have the largest magnitude? Hint 1. Subtracting vectors To subtract two vectors, add a vector with the same magnitude but opposite direction of one of the vectors to the other vector. ANSWER: Correct Tactics Box 3.1 Determining the Components of a Vector Learning Goal: C and D A and F E and F A and B E and D A and F A and E D and B C and D E and F Typesetting math: 15% To practice Tactics Box 3.1 Determining the Components of a Vector. When a vector \texttip{\vec{A}}{A_vec} is decomposed into component vectors \texttip{\vec{A}_{\mit x}}{A_vec_x} and \texttip{\vec{A}_{\mit y}}{A_vec_y} parallel to the coordinate axes, we can describe each component vector with a single number (a scalar) called the component. This tactics box describes how to determine the x component and y component of vector \texttip{\vec{A}}{A_vec}, denoted \texttip{A_{\mit x}}{A_x} and \texttip{A_{\mit y}}{A_y}. TACTICS BOX 3.1 Determining the components of a vector The absolute value |A_x| of the x component \texttip{A_{\mit x}}{A_x} is the magnitude of the component vector \texttip{\vec{A}_{\1. mit x}}{A_vec_x}. The sign of \texttip{A_{\mit x}}{A_x} is positive if \texttip{\vec{A}_{\mit x}}{A_vec_x} points in the positive x direction; it is negative if \texttip{\vec{A}_{\mit x}}{A_vec_x} points in the negative x direction. 2. 3. The y component \texttip{A_{\mit y}}{A_y} is determined similarly. Part A What is the magnitude of the component vector \texttip{\vec{A}_{\mit x}}{A_vec_x} shown in the figure? Express your answer in meters to one significant figure. ANSWER: Correct |A_x| = 5 \rm m Typesetting math: 15% Part B What is the sign of the y component \texttip{A_{\mit y}}{A_y} of vector \texttip{\vec{A}}{A_vec} shown in the figure? ANSWER: Correct Part C Now, combine the information given in the tactics box above to find the x and y components, \texttip{B_{\mit x}}{B_x} and \texttip{B_{\mit y}}{B_y}, of vector \texttip{\vec{B}}{B_vec} shown in the figure. Express your answers, separated by a comma, in meters to one significant figure. positive negative Typesetting math: 15% ANSWER: Correct Conceptual Problem about Projectile Motion Learning Goal: To understand projectile motion by considering horizontal constant velocity motion and vertical constant acceleration motion independently. Projectile motion refers to the motion of unpowered objects (called projectiles) such as balls or stones moving near the surface of the earth under the influence of the earth's gravity alone. In this analysis we assume that air resistance can be neglected. An object undergoing projectile motion near the surface of the earth obeys the following rules: An object undergoing projectile motion travels horizontally at a constant rate. That is, the x component of its velocity, \texttip{v_{\mit x}}{1. v_x}, is constant. An object undergoing projectile motion moves vertically with a constant downward acceleration whose magnitude, denoted by \texttip{g}{g}, is equal to 9.80 \rm{m/s^2} near the surface of the earth. Hence, the y component of its velocity, \texttip{v_{\mit y}}{v_y}, changes continuously. 2. An object undergoing projectile motion will undergo the horizontal and vertical motions described above from the instant it is launched until the instant it strikes the ground again. Even though the horizontal and vertical motions can be treated independently, they are related by the fact that they occur for exactly the same amount of time, namely the time \texttip{t}{t} the projectile is in the air. 3. The figure shows the trajectory (i.e., the path) of a ball undergoing projectile motion over level ground. The time t_0 = 0\;\rm{s} corresponds to the moment just after the ball is launched from position x_0 = 0\;\rm{m} and y_0 = 0\;\rm{m}. Its launch velocity, also called the initial velocity, is \texttip{\vec{v}_{\rm 0}}{v_vec_0}. Two other points along the trajectory are indicated in the figure. One is the moment the ball reaches the peak of its trajectory, at time \texttip{t_{\rm 1}}{t_1} with velocity \texttip{\vec{v}_{\rm 1}}{v_1_vec}. Its position at this moment is denoted by (x_1, y_1) or (x_1, y_{\max}) since it is at its maximum \texttip{B_{\mit x}}{B_x}, \texttip{B_{\mit y}}{B_y} = -2,-5 \rm m, \rm m Typesetting math: 15% The other point, at time \texttip{t_{\rm 2}}{t_2} with velocity \texttip{\vec{v}_{\rm 2}}{v_2_vec}, corresponds to the moment just before the ball strikes the ground on the way back down. At this time its position is (x_2, y_2), also known as (x_{\max}, y_2) since it is at its maximum horizontal range. Projectile motion is symmetric about the peak, provided the object lands at the same vertical height from which is was launched, as is the case here. Hence y_2 = y_0 = 0\;\rm{m}. Part A How do the speeds \texttip{v_{\rm 0}}{v_0}, \texttip{v_{\rm 1}}{v_1}, and \texttip{v_{\rm 2}}{v_2} (at times \texttip{t_{\rm 0}}{t_0}, \texttip{t_{\rm 1}}{t_1}, and \texttip{t_{\rm 2}}{t_2}) compare? ANSWER: Correct Here \texttip{v_{\rm 0}}{v_0} equals \texttip{v_{\rm 2}}{v_2} by symmetry and both exceed \texttip{v_{\rm 1}}{v_1}. This is because \texttip{v_{\rm 0}}{v_0} and \texttip{v_{\rm 2}}{v_2} include vertical speed as well as the constant horizontal speed. Consider a diagram of the ball at time \texttip{t_{\rm 0}}{t_0}. Recall that \texttip{t_{\rm 0}}{t_0} refers to the instant just after the ball has been launched, so it is still at ground level (x_0 = y_0= 0\;\rm{m}). However, it is already moving with initial velocity \texttip{\vec{v}_{\rm 0}}{v_0_vec}, whose magnitude is v_0 = 30.0\;{\rm m/s} and direction is \theta = 60.0\;{\rm degrees} counterclockwise from the positive x direction. \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 1}}{v_1} = \texttip{v_{\rm 2}}{v_2} > 0 \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} = 0 \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} > 0 \texttip{v_{\rm 0}}{v_0} > \texttip{v_{\rm 1}}{v_1} > \texttip{v_{\rm 2}}{v_2} > 0 \texttip{v_{\rm 0}}{v_0} > \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} = 0 Typesetting math: 15% Part B What are the values of the intial velocity vector components \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} (both in \rm{m/s}) as well as the acceleration vector components \texttip{a_{0,x}}{a_0, x} and \texttip{a_{0,y}}{a_0, y} (both in \rm{m/s^2})? Here the subscript 0 means “at time \texttip{t_{\rm 0}}{t_0}.” Hint 1. Determining components of a vector that is aligned with an axis If a vector points along a single axis direction, such as in the positive x direction, its x component will be its full magnitude, whereas its y component will be zero since the vector is perpendicular to the y direction. If the vector points in the negative x direction, its x component will be the negative of its full magnitude. Hint 2. Calculating the components of the initial velocity Notice that the vector \texttip{\vec{v}_{\rm 0}}{v_0_vec} points up and to the right. Since “up” is the positive y axis direction and “to the right” is the positive x axis direction, \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} will both be positive. As shown in the figure, \texttip{v_{0,x}}{v_0, x}, \texttip{v_{0,y}}{v_0, y}, and \texttip{v_{\rm 0}}{v_0} are three sides of a right triangle, one angle of which is \texttip{\theta }{theta}. Thus \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} can be found using the definition of the sine and cosine functions given below. Recall that v_0 = 30.0\;\rm{m/s} and \theta = 60.0\;\rm{degrees} and note that \large{\sin(\theta) = \frac{\rm{length\;of\;opposite\;side}}{\rm{length\;of\;hypotenuse}}} \large{= \frac{v_{0, y}}{v_0}}, \large{\cos(\theta) = \frac{\rm{length\;of\;adjacent\;side}}{\rm{length\;of\;hypotenuse}}} \large{= \frac{v_{0, x}}{v_0}.} What are the values of \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y}? Enter your answers numerically in meters per second separated by a comma. ANSWER: ANSWER: 15.0,26.0 \rm{m/s} Typesetting math: 15% Correct Also notice that at time \texttip{t_{\rm 2}}{t_2}, just before the ball lands, its velocity components are v_{2, x} = 15\;\rm{m/s} (the same as always) and v_{2, y} = – 26.0\;\rm{m/s} (the same size but opposite sign from \texttip{v_{0,y}}{v_0, y} by symmetry). The acceleration at time \texttip{t_{\rm 2}}{t_2} will have components (0, -9.80 \rm{m/s^2}), exactly the same as at \texttip{t_{\rm 0}}{t_0}, as required by Rule 2. The peak of the trajectory occurs at time \texttip{t_{\rm 1}}{t_1}. This is the point where the ball reaches its maximum height \texttip{y_{\rm max}}{y_max}. At the peak the ball switches from moving up to moving down, even as it continues to travel horizontally at a constant rate. Part C What are the values of the velocity vector components \texttip{v_{1,x}}{v_1, x} and \texttip{v_{1,y}}{v_1, y} (both in \rm{m/s}) as well as the acceleration vector components \texttip{a_{1,x}}{a_1, x} and \texttip{a_{1,y}}{a_1, y} (both in \rm{m/s^2})? Here the subscript 1 means that these are all at time \texttip{t_{\rm 1}}{t_1}. ANSWER: 30.0, 0, 0, 0 0, 30.0, 0, 0 15.0, 26.0, 0, 0 30.0, 0, 0, -9.80 0, 30.0, 0, -9.80 15.0, 26.0, 0, -9.80 15.0, 26.0, 0, +9.80 Typesetting math: 15% Correct At the peak of its trajectory the ball continues traveling horizontally at a constant rate. However, at this moment it stops moving up and is about to move back down. This constitutes a downward-directed change in velocity, so the ball is accelerating downward even at the peak. The flight time refers to the total amount of time the ball is in the air, from just after it is launched (\texttip{t_{\rm 0}}{t_0}) until just before it lands (\texttip{t_{\rm 2}}{t_2}). Hence the flight time can be calculated as t_2 – t_0, or just \texttip{t_{\rm 2}}{t_2} in this particular situation since t_0 = 0. Because the ball lands at the same height from which it was launched, by symmetry it spends half its flight time traveling up to the peak and the other half traveling back down. The flight time is determined by the initial vertical component of the velocity and by the acceleration. The flight time does not depend on whether the object is moving horizontally while it is in the air. Part D If a second ball were dropped from rest from height \texttip{y_{\rm max}}{y_max}, how long would it take to reach the ground? Ignore air resistance. Check all that apply. Hint 1. Kicking a ball of cliff; a related problem Consider two balls, one of which is dropped from rest off the edge of a cliff at the same moment that the other is kicked horizontally off the edge of the cliff. Which ball reaches the level ground at the base of the cliff first? Ignore air resistance. Hint 1. Comparing position, velocity, and acceleration of the two balls Both balls start at the same height and have the same initial y velocity (v_{0,y} = 0) as well as the same acceleration (\vec a = g downward). They differ only in their x velocity (one is 0, 0, 0, 0 0, 0, 0, -9.80 15.0, 0, 0, 0 15.0, 0, 0, -9.80 0, 26.0, 0, 0 0, 26.0, 0, -9.80 15.0, 26.0, 0, 0 15.0, 26.0, 0, -9.80 Typesetting math: 15% zero, the other nonzero). This difference will affect their x motion but not their y motion. ANSWER: ANSWER: Correct In projectile motion over level ground, it takes an object just as long to rise from the ground to the peak as it takes for it to fall from the peak back to the ground. The range \texttip{R}{R} of the ball refers to how far it moves horizontally, from just after it is launched until just before it lands. Range is defined as x_2 – x_0, or just \texttip{x_{\rm 2}}{x_2} in this particular situation since x_0 = 0. Range can be calculated as the product of the flight time \texttip{t_{\rm 2}}{t_2} and the x component of the velocity \texttip{v_{\mit x}}{v_x} (which is the same at all times, so v_x = v_{0,x}). The value of \texttip{v_{\mit x}}{v_x} can be found from the launch speed \texttip{v_{\rm 0}}{v_0} and the launch angle \texttip{\theta }{theta} using trigonometric functions, as was done in Part B. The flight time is related to the initial y component of the velocity, which may also be found from \texttip{v_{\rm 0}}{v_0} and \texttip{\theta }{theta} using trig functions. The following equations may be useful in solving projectile motion problems, but these equations apply only to a projectile launched over level ground from position (x_0 = y_0 = 0) at time t_0 = 0 with initial speed \texttip{v_{\rm 0}}{v_0} and launch angle \texttip{\theta }{theta} measured from the horizontal. As was the case above, \texttip{t_{\rm 2}}{t_2} refers to the flight time and \texttip{R}{R} refers to the range of the projectile. flight time: \large{t_2 = \frac{2 v_{0, y}}{g} = \frac{2 v_0 \sin(\theta)}{g}} range: \large{R = v_x t_2 = \frac{v_0^2 \sin(2\theta)}{g}} The ball that falls straight down strikes the ground first. The ball that was kicked so it moves horizontally as it falls strikes the ground first. Both balls strike the ground at the same time. \texttip{t_{\rm 0}}{t_0} t_1 – t_0 \texttip{t_{\rm 2}}{t_2} t_2 – t_1 \large{\frac{t_2 – t_0}{2}} Typesetting math: 15% In general, a high launch angle yields a long flight time but a small horizontal speed and hence little range. A low launch angle gives a larger horizontal speed, but less flight time in which to accumulate range. The launch angle that achieves the maximum range for projectile motion over level ground is 45 degrees. Part E Which of the following changes would increase the range of the ball shown in the original figure? Check all that apply. ANSWER: Correct A solid understanding of the concepts of projectile motion will take you far, including giving you additional insight into the solution of projectile motion problems numerically. Even when the object does not land at the same height from which is was launched, the rules given in the introduction will still be useful. Recall that air resistance is assumed to be negligible here, so this projectile motion analysis may not be the best choice for describing things like frisbees or feathers, whose motion is strongly influenced by air. The value of the gravitational free-fall acceleration \texttip{g}{g} is also assumed to be constant, which may not be appropriate for objects that move vertically through distances of hundreds of kilometers, like rockets or missiles. However, for problems that involve relatively dense projectiles moving close to the surface of the earth, these assumptions are reasonable. A World-Class Sprinter World-class sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has magnitude 15 \;{\rm m}/{\rm s}^{2}. Part A How much horizontal force \texttip{F}{F} must a sprinter of mass 64{\rm kg} exert on the starting blocks to produce this acceleration? Express your answer in newtons using two significant figures. Increase \texttip{v_{\rm 0}}{v_0} above 30 \rm{m/s}. Reduce \texttip{v_{\rm 0}}{v_0} below 30 \rm{m/s}. Reduce \texttip{\theta }{theta} from 60 \rm{degrees} to 45 \rm{degrees}. Reduce \texttip{\theta }{theta} from 60 \rm{degrees} to less than 30 \rm{degrees}. Increase \texttip{\theta }{theta} from 60 \rm{degrees} up toward 90 \rm{degrees}. Typesetting math: 15% Hint 1. Newton’s 2nd law of motion According to Newton’s 2nd law of motion, if a net external force \texttip{F_{\rm net}}{F_net} acts on a body, the body accelerates, and the net force is equal to the mass \texttip{m}{m} of the body times the acceleration \texttip{a}{a} of the body: F_{\rm net} = ma. ANSWER: Co

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Assignment 1 Due: 11:59pm on Wednesday, February 5, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 1.6 Part A Determine the sign (positive or negative) of the position for the particle in the figure. ANSWER: Correct Part B Determine the sign (positive or negative) of the velocity for the particle in the figure. ANSWER: Correct Positive Negative Negative Positive Part C Determine the sign (positive or negative) of the acceleration for the particle in the figure. ANSWER: Correct Conceptual Question 1.7 Part A Determine the sign (positive or negative) of the position for the particle in the figure. ANSWER: Positive Negative Correct Part B Determine the sign (positive or negative) of the velocity for the particle in the figure. ANSWER: Correct Part C Determine the sign (positive or negative) of the acceleration for the particle in the figure. ANSWER: Correct Enhanced EOC: Problem 1.18 The figure shows the motion diagram of a drag racer. The camera took one frame every 2 . Positive Negative Positive Negative Negative Positive s You may want to review ( pages 16 – 19) . For help with math skills, you may want to review: Plotting Points on a Graph Part A Make a position-versus-time graph for the drag racer. Hint 1. How to approach the problem Based on Table 1.1 in the book/e-text, what two observables are associated with each point? Which position or point of the drag racer occurs first? Which position occurs last? If you label the first point as happening at , at what time does the next point occur? At what time does the last position point occur? What is the position of a point halfway in between and ? Can you think of a way to estimate the positions of the points using a ruler? ANSWER: t = 0 s x = 0 m x = 200 m Correct Motion of Two Rockets Learning Goal: To learn to use images of an object in motion to determine velocity and acceleration. Two toy rockets are traveling in the same direction (taken to be the x axis). A diagram is shown of a time-exposure image where a stroboscope has illuminated the rockets at the uniform time intervals indicated. Part A At what time(s) do the rockets have the same velocity? Hint 1. How to determine the velocity The diagram shows position, not velocity. You can’t find instantaneous velocity from this diagram, but you can determine the average velocity between two times and : . Note that no position values are given in the diagram; you will need to estimate these based on the distance between successive positions of the rockets. ANSWER: Correct t1 t2 vavg[t1, t2] = x(t2)−x(t1) t2−t1 at time only at time only at times and at some instant in time between and at no time shown in the figure t = 1 t = 4 t = 1 t = 4 t = 1 t = 4 Part B At what time(s) do the rockets have the same x position? ANSWER: Correct Part C At what time(s) do the two rockets have the same acceleration? Hint 1. How to determine the acceleration The velocity is related to the spacing between images in a stroboscopic diagram. Since acceleration is the rate at which velocity changes, the acceleration is related to the how much this spacing changes from one interval to the next. ANSWER: at time only at time only at times and at some instant in time between and at no time shown in the figure t = 1 t = 4 t = 1 t = 4 t = 1 t = 4 at time only at time only at times and at some instant in time between and at no time shown in the figure t = 1 t = 4 t = 1 t = 4 t = 1 t = 4 Correct Part D The motion of the rocket labeled A is an example of motion with uniform (i.e., constant) __________. ANSWER: Correct Part E The motion of the rocket labeled B is an example of motion with uniform (i.e., constant) __________. ANSWER: Correct Part F At what time(s) is rocket A ahead of rocket B? and nonzero acceleration velocity displacement time and nonzero acceleration velocity displacement time Hint 1. Use the diagram You can answer this question by looking at the diagram and identifying the time(s) when rocket A is to the right of rocket B. ANSWER: Correct Dimensions of Physical Quantities Learning Goal: To introduce the idea of physical dimensions and to learn how to find them. Physical quantities are generally not purely numerical: They have a particular dimension or combination of dimensions associated with them. Thus, your height is not 74, but rather 74 inches, often expressed as 6 feet 2 inches. Although feet and inches are different units they have the same dimension–length. Part A In classical mechanics there are three base dimensions. Length is one of them. What are the other two? Hint 1. MKS system The current system of units is called the International System (abbreviated SI from the French Système International). In the past this system was called the mks system for its base units: meter, kilogram, and second. What are the dimensions of these quantities? ANSWER: before only after only before and after between and at no time(s) shown in the figure t = 1 t = 4 t = 1 t = 4 t = 1 t = 4 Correct There are three dimensions used in mechanics: length ( ), mass ( ), and time ( ). A combination of these three dimensions suffices to express any physical quantity, because when a new physical quantity is needed (e.g., velocity), it always obeys an equation that permits it to be expressed in terms of the units used for these three dimensions. One then derives a unit to measure the new physical quantity from that equation, and often its unit is given a special name. Such new dimensions are called derived dimensions and the units they are measured in are called derived units. For example, area has derived dimensions . (Note that “dimensions of variable ” is symbolized as .) You can find these dimensions by looking at the formula for the area of a square , where is the length of a side of the square. Clearly . Plugging this into the equation gives . Part B Find the dimensions of volume. Express your answer as powers of length ( ), mass ( ), and time ( ). Hint 1. Equation for volume You have likely learned many formulas for the volume of various shapes in geometry. Any of these equations will give you the dimensions for volume. You can find the dimensions most easily from the volume of a cube , where is the length of the edge of the cube. ANSWER: acceleration and mass acceleration and time acceleration and charge mass and time mass and charge time and charge l m t A [A] = l2 x [x] A = s2 s [s] = l [A] = [s] = 2 l2 [V ] l m t V = e3 e [V ] = l3 Correct Part C Find the dimensions of speed. Express your answer as powers of length ( ), mass ( ), and time ( ). Hint 1. Equation for speed Speed is defined in terms of distance and time as . Therefore, . Hint 2. Familiar units for speed You are probably accustomed to hearing speeds in miles per hour (or possibly kilometers per hour). Think about the dimensions for miles and hours. If you divide the dimensions for miles by the dimensions for hours, you will have the dimensions for speed. ANSWER: Correct The dimensions of a quantity are not changed by addition or subtraction of another quantity with the same dimensions. This means that , which comes from subtracting two speeds, has the same dimensions as speed. It does not make physical sense to add or subtract two quanitites that have different dimensions, like length plus time. You can add quantities that have different units, like miles per hour and kilometers per hour, as long as you convert both quantities to the same set of units before you actually compute the sum. You can use this rule to check your answers to any physics problem you work. If the answer involves the sum or difference of two quantities with different dimensions, then it must be incorrect. This rule also ensures that the dimensions of any physical quantity will never involve sums or differences of the base dimensions. (As in the preceeding example, is not a valid dimension for a [v] l m t v d t v = d t [v] = [d]/[t] [v] = lt−1 v l + t physical quantitiy.) A valid dimension will only involve the product or ratio of powers of the base dimensions (e.g. ). Part D Find the dimensions of acceleration. Express your answer as powers of length ( ), mass ( ), and time ( ). Hint 1. Equation for acceleration In physics, acceleration is defined as the change in velocity in a certain time. This is shown by the equation . The is a symbol that means “the change in.” ANSWER: Correct Consistency of Units In physics, every physical quantity is measured with respect to a unit. Time is measured in seconds, length is measured in meters, and mass is measured in kilograms. Knowing the units of physical quantities will help you solve problems in physics. Part A Gravity causes objects to be attracted to one another. This attraction keeps our feet firmly planted on the ground and causes the moon to orbit the earth. The force of gravitational attraction is represented by the equation , where is the magnitude of the gravitational attraction on either body, and are the masses of the bodies, is the distance between them, and is the gravitational constant. In SI units, the units of force are , the units of mass are , and the units of distance are . For this equation to have consistent units, the units of must be which of the following? Hint 1. How to approach the problem To solve this problem, we start with the equation m2/3 l2 t−2 [a] l m t a a = v/t  [a] = lt−2 F = Gm1m2 r2 F m1 m2 r G kg  m/s2 kg m G . For each symbol whose units we know, we replace the symbol with those units. For example, we replace with . We now solve this equation for . ANSWER: Correct Part B One consequence of Einstein’s theory of special relativity is that mass is a form of energy. This mass-energy relationship is perhaps the most famous of all physics equations: , where is mass, is the speed of the light, and is the energy. In SI units, the units of speed are . For the preceding equation to have consistent units (the same units on both sides of the equation), the units of must be which of the following? Hint 1. How to approach the problem To solve this problem, we start with the equation . For each symbol whose units we know, we replace the symbol with those units. For example, we replace with . We now solve this equation for . ANSWER: F = Gm1m2 r2 m1 kg G kg3 ms2 kgs2 m3 m3 kgs2 m kgs2 E = mc2 m c E m/s E E = mc2 m kg E Correct To solve the types of problems typified by these examples, we start with the given equation. For each symbol whose units we know, we replace the symbol with those units. For example, we replace with . We now solve this equation for the units of the unknown variable. Problem 1.24 Convert the following to SI units: Part A 5.0 Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B 54 Express your answer to two significant figures and include the appropriate units. kgm s kgm2 s2 kgs2 m2 kgm2 s m kg in 0.13 m ft/s ANSWER: Correct Part C 72 Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D 17 Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 1.55 The figure shows a motion diagram of a car traveling down a street. The camera took one frame every 10 . A distance scale is provided. 16 ms mph 32 ms in2 1.1×10−2 m2 s Part A Make a position-versus-time graph for the car. ANSWER: Incorrect; Try Again ± Moving at the Speed of Light Part A How many nanoseconds does it take light to travel a distance of 4.40 in vacuum? Express your answer numerically in nanoseconds. Hint 1. How to approach the problem Light travels at a constant speed; therefore, you can use the formula for the distance traveled in a certain amount of time by an object moving at constant speed. Before performing any calculations, it is often recommended, although it is not strictly necessary, to convert all quantities to their fundamental units rather than to multiples of the fundamental unit. km Hint 2. Find how many seconds it takes light to travel the given distance Given that the speed of light in vacuum is , how many seconds does it take light to travel a distance of 4.40 ? Express your answer numerically in seconds. Hint 1. Find the time it takes light to travel a certain distance How long does it take light to travel a distance ? Let be the speed of light. Hint 1. The speed of an object The equation that relates the distance traveled by an object with constant speed in a time is . ANSWER: Correct Hint 2. Convert the given distance to meters Convert = 4.40 to meters. Express your answer numerically in meters. Hint 1. Conversion of kilometers to meters Recall that . 3.00 × 108 m/s km r c s v t s = vt r  c r c c r d km 1 km = 103 m ANSWER: Correct ANSWER: Correct Now convert the time into nanoseconds. Recall that . ANSWER: Correct Score Summary: Your score on this assignment is 84.7%. You received 50.84 out of a possible total of 60 points. 4.40km = 4400 m 1.47×10−5 s 1 ns = 10−9 s 1.47×104 ns

Assignment 1 Due: 11:59pm on Wednesday, February 5, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 1.6 Part A Determine the sign (positive or negative) of the position for the particle in the figure. ANSWER: Correct Part B Determine the sign (positive or negative) of the velocity for the particle in the figure. ANSWER: Correct Positive Negative Negative Positive Part C Determine the sign (positive or negative) of the acceleration for the particle in the figure. ANSWER: Correct Conceptual Question 1.7 Part A Determine the sign (positive or negative) of the position for the particle in the figure. ANSWER: Positive Negative Correct Part B Determine the sign (positive or negative) of the velocity for the particle in the figure. ANSWER: Correct Part C Determine the sign (positive or negative) of the acceleration for the particle in the figure. ANSWER: Correct Enhanced EOC: Problem 1.18 The figure shows the motion diagram of a drag racer. The camera took one frame every 2 . Positive Negative Positive Negative Negative Positive s You may want to review ( pages 16 – 19) . For help with math skills, you may want to review: Plotting Points on a Graph Part A Make a position-versus-time graph for the drag racer. Hint 1. How to approach the problem Based on Table 1.1 in the book/e-text, what two observables are associated with each point? Which position or point of the drag racer occurs first? Which position occurs last? If you label the first point as happening at , at what time does the next point occur? At what time does the last position point occur? What is the position of a point halfway in between and ? Can you think of a way to estimate the positions of the points using a ruler? ANSWER: t = 0 s x = 0 m x = 200 m Correct Motion of Two Rockets Learning Goal: To learn to use images of an object in motion to determine velocity and acceleration. Two toy rockets are traveling in the same direction (taken to be the x axis). A diagram is shown of a time-exposure image where a stroboscope has illuminated the rockets at the uniform time intervals indicated. Part A At what time(s) do the rockets have the same velocity? Hint 1. How to determine the velocity The diagram shows position, not velocity. You can’t find instantaneous velocity from this diagram, but you can determine the average velocity between two times and : . Note that no position values are given in the diagram; you will need to estimate these based on the distance between successive positions of the rockets. ANSWER: Correct t1 t2 vavg[t1, t2] = x(t2)−x(t1) t2−t1 at time only at time only at times and at some instant in time between and at no time shown in the figure t = 1 t = 4 t = 1 t = 4 t = 1 t = 4 Part B At what time(s) do the rockets have the same x position? ANSWER: Correct Part C At what time(s) do the two rockets have the same acceleration? Hint 1. How to determine the acceleration The velocity is related to the spacing between images in a stroboscopic diagram. Since acceleration is the rate at which velocity changes, the acceleration is related to the how much this spacing changes from one interval to the next. ANSWER: at time only at time only at times and at some instant in time between and at no time shown in the figure t = 1 t = 4 t = 1 t = 4 t = 1 t = 4 at time only at time only at times and at some instant in time between and at no time shown in the figure t = 1 t = 4 t = 1 t = 4 t = 1 t = 4 Correct Part D The motion of the rocket labeled A is an example of motion with uniform (i.e., constant) __________. ANSWER: Correct Part E The motion of the rocket labeled B is an example of motion with uniform (i.e., constant) __________. ANSWER: Correct Part F At what time(s) is rocket A ahead of rocket B? and nonzero acceleration velocity displacement time and nonzero acceleration velocity displacement time Hint 1. Use the diagram You can answer this question by looking at the diagram and identifying the time(s) when rocket A is to the right of rocket B. ANSWER: Correct Dimensions of Physical Quantities Learning Goal: To introduce the idea of physical dimensions and to learn how to find them. Physical quantities are generally not purely numerical: They have a particular dimension or combination of dimensions associated with them. Thus, your height is not 74, but rather 74 inches, often expressed as 6 feet 2 inches. Although feet and inches are different units they have the same dimension–length. Part A In classical mechanics there are three base dimensions. Length is one of them. What are the other two? Hint 1. MKS system The current system of units is called the International System (abbreviated SI from the French Système International). In the past this system was called the mks system for its base units: meter, kilogram, and second. What are the dimensions of these quantities? ANSWER: before only after only before and after between and at no time(s) shown in the figure t = 1 t = 4 t = 1 t = 4 t = 1 t = 4 Correct There are three dimensions used in mechanics: length ( ), mass ( ), and time ( ). A combination of these three dimensions suffices to express any physical quantity, because when a new physical quantity is needed (e.g., velocity), it always obeys an equation that permits it to be expressed in terms of the units used for these three dimensions. One then derives a unit to measure the new physical quantity from that equation, and often its unit is given a special name. Such new dimensions are called derived dimensions and the units they are measured in are called derived units. For example, area has derived dimensions . (Note that “dimensions of variable ” is symbolized as .) You can find these dimensions by looking at the formula for the area of a square , where is the length of a side of the square. Clearly . Plugging this into the equation gives . Part B Find the dimensions of volume. Express your answer as powers of length ( ), mass ( ), and time ( ). Hint 1. Equation for volume You have likely learned many formulas for the volume of various shapes in geometry. Any of these equations will give you the dimensions for volume. You can find the dimensions most easily from the volume of a cube , where is the length of the edge of the cube. ANSWER: acceleration and mass acceleration and time acceleration and charge mass and time mass and charge time and charge l m t A [A] = l2 x [x] A = s2 s [s] = l [A] = [s] = 2 l2 [V ] l m t V = e3 e [V ] = l3 Correct Part C Find the dimensions of speed. Express your answer as powers of length ( ), mass ( ), and time ( ). Hint 1. Equation for speed Speed is defined in terms of distance and time as . Therefore, . Hint 2. Familiar units for speed You are probably accustomed to hearing speeds in miles per hour (or possibly kilometers per hour). Think about the dimensions for miles and hours. If you divide the dimensions for miles by the dimensions for hours, you will have the dimensions for speed. ANSWER: Correct The dimensions of a quantity are not changed by addition or subtraction of another quantity with the same dimensions. This means that , which comes from subtracting two speeds, has the same dimensions as speed. It does not make physical sense to add or subtract two quanitites that have different dimensions, like length plus time. You can add quantities that have different units, like miles per hour and kilometers per hour, as long as you convert both quantities to the same set of units before you actually compute the sum. You can use this rule to check your answers to any physics problem you work. If the answer involves the sum or difference of two quantities with different dimensions, then it must be incorrect. This rule also ensures that the dimensions of any physical quantity will never involve sums or differences of the base dimensions. (As in the preceeding example, is not a valid dimension for a [v] l m t v d t v = d t [v] = [d]/[t] [v] = lt−1 v l + t physical quantitiy.) A valid dimension will only involve the product or ratio of powers of the base dimensions (e.g. ). Part D Find the dimensions of acceleration. Express your answer as powers of length ( ), mass ( ), and time ( ). Hint 1. Equation for acceleration In physics, acceleration is defined as the change in velocity in a certain time. This is shown by the equation . The is a symbol that means “the change in.” ANSWER: Correct Consistency of Units In physics, every physical quantity is measured with respect to a unit. Time is measured in seconds, length is measured in meters, and mass is measured in kilograms. Knowing the units of physical quantities will help you solve problems in physics. Part A Gravity causes objects to be attracted to one another. This attraction keeps our feet firmly planted on the ground and causes the moon to orbit the earth. The force of gravitational attraction is represented by the equation , where is the magnitude of the gravitational attraction on either body, and are the masses of the bodies, is the distance between them, and is the gravitational constant. In SI units, the units of force are , the units of mass are , and the units of distance are . For this equation to have consistent units, the units of must be which of the following? Hint 1. How to approach the problem To solve this problem, we start with the equation m2/3 l2 t−2 [a] l m t a a = v/t  [a] = lt−2 F = Gm1m2 r2 F m1 m2 r G kg  m/s2 kg m G . For each symbol whose units we know, we replace the symbol with those units. For example, we replace with . We now solve this equation for . ANSWER: Correct Part B One consequence of Einstein’s theory of special relativity is that mass is a form of energy. This mass-energy relationship is perhaps the most famous of all physics equations: , where is mass, is the speed of the light, and is the energy. In SI units, the units of speed are . For the preceding equation to have consistent units (the same units on both sides of the equation), the units of must be which of the following? Hint 1. How to approach the problem To solve this problem, we start with the equation . For each symbol whose units we know, we replace the symbol with those units. For example, we replace with . We now solve this equation for . ANSWER: F = Gm1m2 r2 m1 kg G kg3 ms2 kgs2 m3 m3 kgs2 m kgs2 E = mc2 m c E m/s E E = mc2 m kg E Correct To solve the types of problems typified by these examples, we start with the given equation. For each symbol whose units we know, we replace the symbol with those units. For example, we replace with . We now solve this equation for the units of the unknown variable. Problem 1.24 Convert the following to SI units: Part A 5.0 Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B 54 Express your answer to two significant figures and include the appropriate units. kgm s kgm2 s2 kgs2 m2 kgm2 s m kg in 0.13 m ft/s ANSWER: Correct Part C 72 Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D 17 Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 1.55 The figure shows a motion diagram of a car traveling down a street. The camera took one frame every 10 . A distance scale is provided. 16 ms mph 32 ms in2 1.1×10−2 m2 s Part A Make a position-versus-time graph for the car. ANSWER: Incorrect; Try Again ± Moving at the Speed of Light Part A How many nanoseconds does it take light to travel a distance of 4.40 in vacuum? Express your answer numerically in nanoseconds. Hint 1. How to approach the problem Light travels at a constant speed; therefore, you can use the formula for the distance traveled in a certain amount of time by an object moving at constant speed. Before performing any calculations, it is often recommended, although it is not strictly necessary, to convert all quantities to their fundamental units rather than to multiples of the fundamental unit. km Hint 2. Find how many seconds it takes light to travel the given distance Given that the speed of light in vacuum is , how many seconds does it take light to travel a distance of 4.40 ? Express your answer numerically in seconds. Hint 1. Find the time it takes light to travel a certain distance How long does it take light to travel a distance ? Let be the speed of light. Hint 1. The speed of an object The equation that relates the distance traveled by an object with constant speed in a time is . ANSWER: Correct Hint 2. Convert the given distance to meters Convert = 4.40 to meters. Express your answer numerically in meters. Hint 1. Conversion of kilometers to meters Recall that . 3.00 × 108 m/s km r c s v t s = vt r  c r c c r d km 1 km = 103 m ANSWER: Correct ANSWER: Correct Now convert the time into nanoseconds. Recall that . ANSWER: Correct Score Summary: Your score on this assignment is 84.7%. You received 50.84 out of a possible total of 60 points. 4.40km = 4400 m 1.47×10−5 s 1 ns = 10−9 s 1.47×104 ns

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A diver of mass m has an angular velocity w1 when he leaves the diving board, at which time he is stretched out straight. In this position, a reasonably fair model is to treat his body as a uniform rod with a length of 1.5 meters rotating about its midpoint. He then goes into a “cannon ball” position, which can be modeled as cylinder with radius of 0.25 meters.

A diver of mass m has an angular velocity w1 when he leaves the diving board, at which time he is stretched out straight. In this position, a reasonably fair model is to treat his body as a uniform rod with a length of 1.5 meters rotating about its midpoint. He then goes into a “cannon ball” position, which can be modeled as cylinder with radius of 0.25 meters.

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Problems Marking scheme 1. Let A be a nonzero square matrix. Is it possible that a positive integer k exists such that ?? = 0 ? For example, find ?3 for the matrix [ 0 1 2 0 0 1 0 0 0 ] A square matrix A is nilpotent of index k when ? ≠ 0 , ?2 ≠ 0 , … . . , ??−1 ≠ 0, ??? ?? = 0. In this task you will explore nilpotent matrices. 1. The matrix in the example given above is nilpotent. What is its index? ( 2 marks ) 2. Use a software program to determine which of the following matrices are nilpotent and find their indices ( 12 marks ) A. [ 0 1 0 0 ] B. [ 0 1 1 0 ] C. [ 0 0 1 0 ] D. [ 1 0 1 0 ] E. [ 0 0 1 0 0 0 0 0 0 ] F. [ 0 0 0 1 0 0 1 1 0 ] 3. Find 3×3 nilpotent matrices of indices 2 and 3 ( 2 marks ) 4. Find 4×4 nilpotent matrices of indices 2, 3, and 4 ( 2 marks ) 5. Find nilpotent matrix of index 5 ( 2 marks ) 6. Are nilpotent matrices invertible? prove your answer ( 3 marks ) 7. When A is nilpotent, what can you say about ?? ? prove your answer ( 3 marks ) 8. Show that if ? is nilpotent , then ? − ? is invertible ( 4 marks ) 30% 2. A radio transmitter circuit contains a resisitance of 2.0 Ω, a variable inductor of 100 − ? ℎ?????? and a voltage source of 4.0 ? . find the current ? in the circuit as a function of the time t for 0 ≤ ? ≤ 100? if the intial curent is zero. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 3. An object falling under the influence of gravity has a variable accelertaion given by 32 − ? , where ? represents the velocity. If the object starts from rest, find an expression for the velocity in terms of the time. Also, find the limiting value of the velocity. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 4. When the angular displacement ? of a pendulum is small ( less than 60), the pendulum moves with simple harmonic motion closely approximated by ?′′ + ? ? ? = 0 . Here , ?′ = ?? ?? and ? is the accelertaion due to gravity , and ? is the length of the pendulum. Find ? as a function of time ( in s ) if ? = 9.8 ?/?2, ? = 1.0 ? ? = 0.1 and ?? ?? = 0 when ? = 0 . sketch the cuve using any graphical tool. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 5. Find the equation relating the charge and the time in a electric circuit with the following elements: ? = 0.200 ? , ? = 8.00 Ω , ? = 1.00 ?? , ? = 0. In this circuit , ? = 0 and ? = 0.500 ? when ? = 0 Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 6. A spring is stretched 1 m by ? 20 − ? Weight. The spring is stretched 0.5 m below the equilibrium position with the weight attached and the then released. If it is a medium that resists the motion with a force equal to 12?, where v is the velocity, sketch and find the displacement y of the weight as a function of the time. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 7. A 20?? inductor, a 40.0 Ω resistor, a 50.0 ?? capacitor, and voltage source of 100 ?−100?are connected in series in an electric circuit. Find the charge on the capacitor as a function of time t , if ? = 0 and ? = 0 ?ℎ?? ? = 0 Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 10% quality and neatness and using Math equations in MS word. –

Problems Marking scheme 1. Let A be a nonzero square matrix. Is it possible that a positive integer k exists such that ?? = 0 ? For example, find ?3 for the matrix [ 0 1 2 0 0 1 0 0 0 ] A square matrix A is nilpotent of index k when ? ≠ 0 , ?2 ≠ 0 , … . . , ??−1 ≠ 0, ??? ?? = 0. In this task you will explore nilpotent matrices. 1. The matrix in the example given above is nilpotent. What is its index? ( 2 marks ) 2. Use a software program to determine which of the following matrices are nilpotent and find their indices ( 12 marks ) A. [ 0 1 0 0 ] B. [ 0 1 1 0 ] C. [ 0 0 1 0 ] D. [ 1 0 1 0 ] E. [ 0 0 1 0 0 0 0 0 0 ] F. [ 0 0 0 1 0 0 1 1 0 ] 3. Find 3×3 nilpotent matrices of indices 2 and 3 ( 2 marks ) 4. Find 4×4 nilpotent matrices of indices 2, 3, and 4 ( 2 marks ) 5. Find nilpotent matrix of index 5 ( 2 marks ) 6. Are nilpotent matrices invertible? prove your answer ( 3 marks ) 7. When A is nilpotent, what can you say about ?? ? prove your answer ( 3 marks ) 8. Show that if ? is nilpotent , then ? − ? is invertible ( 4 marks ) 30% 2. A radio transmitter circuit contains a resisitance of 2.0 Ω, a variable inductor of 100 − ? ℎ?????? and a voltage source of 4.0 ? . find the current ? in the circuit as a function of the time t for 0 ≤ ? ≤ 100? if the intial curent is zero. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 3. An object falling under the influence of gravity has a variable accelertaion given by 32 − ? , where ? represents the velocity. If the object starts from rest, find an expression for the velocity in terms of the time. Also, find the limiting value of the velocity. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 4. When the angular displacement ? of a pendulum is small ( less than 60), the pendulum moves with simple harmonic motion closely approximated by ?′′ + ? ? ? = 0 . Here , ?′ = ?? ?? and ? is the accelertaion due to gravity , and ? is the length of the pendulum. Find ? as a function of time ( in s ) if ? = 9.8 ?/?2, ? = 1.0 ? ? = 0.1 and ?? ?? = 0 when ? = 0 . sketch the cuve using any graphical tool. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 5. Find the equation relating the charge and the time in a electric circuit with the following elements: ? = 0.200 ? , ? = 8.00 Ω , ? = 1.00 ?? , ? = 0. In this circuit , ? = 0 and ? = 0.500 ? when ? = 0 Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 6. A spring is stretched 1 m by ? 20 − ? Weight. The spring is stretched 0.5 m below the equilibrium position with the weight attached and the then released. If it is a medium that resists the motion with a force equal to 12?, where v is the velocity, sketch and find the displacement y of the weight as a function of the time. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 7. A 20?? inductor, a 40.0 Ω resistor, a 50.0 ?? capacitor, and voltage source of 100 ?−100?are connected in series in an electric circuit. Find the charge on the capacitor as a function of time t , if ? = 0 and ? = 0 ?ℎ?? ? = 0 Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 10% quality and neatness and using Math equations in MS word. –

Problems Marking scheme 1. Let A be a nonzero square … Read More...
Atomic radius generally increases as we move __________. A) down a group and from right to left across a period B) up a group and from left to right across a period C) down a group and from left to right across a period D) up a group and from right to left across a period E) down a group; the period position has no effect

Atomic radius generally increases as we move __________. A) down a group and from right to left across a period B) up a group and from left to right across a period C) down a group and from left to right across a period D) up a group and from right to left across a period E) down a group; the period position has no effect

A) down a group and from right to left across … Read More...
You will receive no credit for items you complete after the assignment is due. Grading Policy Exercise 2.5 Starting from the front door of your ranch house, you walk 60.0 due east to your windmill, and then you turn around and slowly walk 35.0 west to a bench where you sit and watch the sunrise. It takes you 27.0 to walk from your house to the windmill and then 49.0 to walk from the windmill to the bench. Part A For the entire trip from your front door to the bench, what is your average velocity? Express your answer with the appropriate units. ANSWER: Correct Part B For the entire trip from your front door to the bench, what is your average speed? Express your answer with the appropriate units. ANSWER: Correct Exercise 2.7 A car is stopped at a traffic light. It then travels along a straight road so that its distance from the light is given by , where = 2.40 and = 0.110 . = -0.329 average speed = 1.25 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 1 of 16 3/23/2015 11:12 AM Part A Calculate the average velocity of the car for the time interval = 0 to = 10.0 . ANSWER: Correct Part B Calculate the instantaneous velocity of the car at =0. ANSWER: Correct Part C Calculate the instantaneous velocity of the car at =5.00 . ANSWER: Correct Part D Calculate the instantaneous velocity of the car at =10.0 . ANSWER: Correct Part E How long after starting from rest is the car again at rest? ANSWER: = 13.0 = 0 = 15.8 = 15.0 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 2 of 16 3/23/2015 11:12 AM Correct Exercise 2.9 A ball moves in a straight line (the x-axis). The graph in the figure shows this ball’s velocity as a function of time. Part A What are the ball’s average velocity during the first 2.8 ? Express your answer using two significant figures. ANSWER: Answer Requested Part B What are the ball’s average speed during the first 2.8 ? Express your answer using two significant figures. ANSWER: Correct = 14.5 = 2.3 = 2.3 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 3 of 16 3/23/2015 11:12 AM Part C Suppose that the ball moved in such a way that the graph segment after 2.0 was -3.0 instead of +3.0 . Find the ball’s and average velocity during the first 2.8 in this case. Express your answer using two significant figures. ANSWER: All attempts used; correct answer displayed Part D Suppose that the ball moved in such a way that the graph segment after 2.0 was -3.0 instead of +3.0 . Find the ball’s average speed during the first 2.8 in this case. Express your answer using two significant figures. ANSWER: Correct Exercise 2.13 Part A The table shows test data for the Bugatti Veyron, the fastest car made. The car is moving in a straight line (the x-axis). Time 0 2.10 20.0 53.0 Speed 0 60.0 205 259 Calculate the car’s average acceleration (in ) between 0 and 2.1 . ANSWER: Correct Part B Calculate the car’s average acceleration (in ) between 2.1 and 20.0 . = 0.57 = 2.3 = 12.8 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 4 of 16 3/23/2015 11:12 AM ANSWER: Correct Part C Calculate the car’s average acceleration (in ) between 20.0 and 53 . ANSWER: Correct Exercise 2.19 An antelope moving with constant acceleration covers the distance 79.0 between two points in time 7.00 . Its speed as it passes the second point is 14.5 . Part A What is its speed at the first point? ANSWER: Correct Part B What is the acceleration? ANSWER: Correct Exercise 2.22 In the fastest measured tennis serve, the ball left the racquet at 73.14 . A served tennis ball is typically in contact with = 3.62 = 0.731 = 8.07 = 0.918 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 5 of 16 3/23/2015 11:12 AM the racquet for 27.0 and starts from rest. Assume constant acceleration. Part A What was the ball’s acceleration during this serve? ANSWER: Correct Part B How far did the ball travel during the serve? ANSWER: Correct Exercise 2.30 A cat walks in a straight line, which we shall call the x-axis with the positive direction to the right. As an observant physicist, you make measurements of this cat’s motion and construct a graph of the feline’s velocity as a function of time (the figure ). Part A Find the cat’s velocity at = 5.0 . Express your answer using two significant figures. ANSWER: = 2710 = 0.987 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 6 of 16 3/23/2015 11:12 AM Correct Part B Find the cat’s velocity at = 8.0 . Express your answer using two significant figures. ANSWER: Correct Part C What is the cat’s acceleration at ? Express your answer using two significant figures. ANSWER: Correct Part D What is the cat’s acceleration at ? Express your answer using two significant figures. ANSWER: Correct Part E What is the cat’s acceleration at ? Express your answer using two significant figures. ANSWER: = 1.3 = -2.7 = -1.3 = -1.3 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 7 of 16 3/23/2015 11:12 AM Correct Part F What distance does the cat move during the first 4.5 ? Express your answer using two significant figures. ANSWER: Correct Part G What distance does the cat move from to ? Express your answer using two significant figures. ANSWER: Correct Part H Sketch clear graph of the cat’s acceleration as function of time, assuming that the cat started at the origin. ANSWER: = -1.3 = 23 = 26 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 8 of 16 3/23/2015 11:12 AM Correct Part I Sketch clear graph of the cat’s position as function of time, assuming that the cat started at the origin. ANSWER: Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 9 of 16 3/23/2015 11:12 AM All attempts used; correct answer displayed Exercise 2.35 Part A If a flea can jump straight up to a height of 0.510 , what is its initial speed as it leaves the ground? ANSWER: Correct Part B How long is it in the air? ANSWER: Correct = 3.16 = 0.645 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 10 of 16 3/23/2015 11:12 AM Exercise 2.36 A small rock is thrown vertically upward with a speed of 18.0 from the edge of the roof of a 39.0 tall building. The rock doesn’t hit the building on its way back down and lands in the street below. Air resistance can be neglected. Part A What is the speed of the rock just before it hits the street? Express your answer with the appropriate units. ANSWER: Correct Part B How much time elapses from when the rock is thrown until it hits the street? Express your answer with the appropriate units. ANSWER: Correct Exercise 2.38 You throw a glob of putty straight up toward the ceiling, which is 3.00 above the point where the putty leaves your hand. The initial speed of the putty as it leaves your hand is 9.70 . Part A What is the speed of the putty just before it strikes the ceiling? Express your answer with the appropriate units. ANSWER: Correct Part B = 33.0 = 5.20 = 5.94 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 11 of 16 3/23/2015 11:12 AM How much time from when it leaves your hand does it take the putty to reach the ceiling? Express your answer with the appropriate units. ANSWER: Correct Exercise 3.1 A squirrel has x- and y-coordinates ( 1.2 , 3.3 ) at time and coordinates ( 5.3 , -0.80 ) at time = 2.6 . Part A For this time interval, find the x-component of the average velocity. Express your answer using two significant figures. ANSWER: Correct Part B For this time interval, find the y-component of the average velocity. Express your answer using two significant figures. ANSWER: Correct Part C Find the magnitude of the average velocity. Express your answer using two significant figures. ANSWER: = 0.384 = 1.6 = -1.6 = 2.2 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 12 of 16 3/23/2015 11:12 AM Correct Part D Find the direction of the average velocity. Express your answer using two significant figures. ANSWER: Correct Exercise 3.3 A web page designer creates an animation in which a dot on a computer screen has a position of 4.1 2.1 4.7 . Part A Find the average velocity of the dot between and . Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Part B Find the instantaneous velocity at . Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Part C = 45 below the x-axis = 4.2,4.7 = 0,4.7 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 13 of 16 3/23/2015 11:12 AM Find the instantaneous velocity at . Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Part D Find the instantaneous velocity at . Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Exercise 3.5 A jet plane is flying at a constant altitude. At time it has components of velocity 89 , 108 . At time 32.5 the components are 165 , 37 . Part A For this time interval calculate the average acceleration. Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Part B Find the magnitude of the average acceleration. Express your answer using two significant figures. ANSWER: = 4.2,4.7 = 8.4,4.7 = 2.3,-2.2 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 14 of 16 3/23/2015 11:12 AM Correct Part C Find the direction of the average acceleration (let the direction be the angle that the vector makes with the +x-axis, measured counterclockwise). ANSWER: Correct Exercise 3.4 The position of a squirrel running in a park is given by . Part A What is , the -component of the velocity of the squirrel, as function of time? ANSWER: Correct Part B What is , the y-component of the velocity of the squirrel, as function of time? ANSWER: = 3.2 = -43.1 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 15 of 16 3/23/2015 11:12 AM Correct Part C At 4.51 , how far is the squirrel from its initial position? Express your answer to three significant figures and include the appropriate units. ANSWER: All attempts used; correct answer displayed Part D At 4.51 , what is the magnitude of the squirrel’s velocity? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part E At 4.51 , what is the direction (in degrees counterclockwise from +x-axis) of the squirrel’s velocity? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Score Summary: Your score on this assignment is 90.1%. You received 14.42 out of a possible total of 16 points. = 2.65 = 1.31 = 62.5 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 16 of 16 3/23/2015 11:12 AM

You will receive no credit for items you complete after the assignment is due. Grading Policy Exercise 2.5 Starting from the front door of your ranch house, you walk 60.0 due east to your windmill, and then you turn around and slowly walk 35.0 west to a bench where you sit and watch the sunrise. It takes you 27.0 to walk from your house to the windmill and then 49.0 to walk from the windmill to the bench. Part A For the entire trip from your front door to the bench, what is your average velocity? Express your answer with the appropriate units. ANSWER: Correct Part B For the entire trip from your front door to the bench, what is your average speed? Express your answer with the appropriate units. ANSWER: Correct Exercise 2.7 A car is stopped at a traffic light. It then travels along a straight road so that its distance from the light is given by , where = 2.40 and = 0.110 . = -0.329 average speed = 1.25 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 1 of 16 3/23/2015 11:12 AM Part A Calculate the average velocity of the car for the time interval = 0 to = 10.0 . ANSWER: Correct Part B Calculate the instantaneous velocity of the car at =0. ANSWER: Correct Part C Calculate the instantaneous velocity of the car at =5.00 . ANSWER: Correct Part D Calculate the instantaneous velocity of the car at =10.0 . ANSWER: Correct Part E How long after starting from rest is the car again at rest? ANSWER: = 13.0 = 0 = 15.8 = 15.0 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 2 of 16 3/23/2015 11:12 AM Correct Exercise 2.9 A ball moves in a straight line (the x-axis). The graph in the figure shows this ball’s velocity as a function of time. Part A What are the ball’s average velocity during the first 2.8 ? Express your answer using two significant figures. ANSWER: Answer Requested Part B What are the ball’s average speed during the first 2.8 ? Express your answer using two significant figures. ANSWER: Correct = 14.5 = 2.3 = 2.3 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 3 of 16 3/23/2015 11:12 AM Part C Suppose that the ball moved in such a way that the graph segment after 2.0 was -3.0 instead of +3.0 . Find the ball’s and average velocity during the first 2.8 in this case. Express your answer using two significant figures. ANSWER: All attempts used; correct answer displayed Part D Suppose that the ball moved in such a way that the graph segment after 2.0 was -3.0 instead of +3.0 . Find the ball’s average speed during the first 2.8 in this case. Express your answer using two significant figures. ANSWER: Correct Exercise 2.13 Part A The table shows test data for the Bugatti Veyron, the fastest car made. The car is moving in a straight line (the x-axis). Time 0 2.10 20.0 53.0 Speed 0 60.0 205 259 Calculate the car’s average acceleration (in ) between 0 and 2.1 . ANSWER: Correct Part B Calculate the car’s average acceleration (in ) between 2.1 and 20.0 . = 0.57 = 2.3 = 12.8 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 4 of 16 3/23/2015 11:12 AM ANSWER: Correct Part C Calculate the car’s average acceleration (in ) between 20.0 and 53 . ANSWER: Correct Exercise 2.19 An antelope moving with constant acceleration covers the distance 79.0 between two points in time 7.00 . Its speed as it passes the second point is 14.5 . Part A What is its speed at the first point? ANSWER: Correct Part B What is the acceleration? ANSWER: Correct Exercise 2.22 In the fastest measured tennis serve, the ball left the racquet at 73.14 . A served tennis ball is typically in contact with = 3.62 = 0.731 = 8.07 = 0.918 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 5 of 16 3/23/2015 11:12 AM the racquet for 27.0 and starts from rest. Assume constant acceleration. Part A What was the ball’s acceleration during this serve? ANSWER: Correct Part B How far did the ball travel during the serve? ANSWER: Correct Exercise 2.30 A cat walks in a straight line, which we shall call the x-axis with the positive direction to the right. As an observant physicist, you make measurements of this cat’s motion and construct a graph of the feline’s velocity as a function of time (the figure ). Part A Find the cat’s velocity at = 5.0 . Express your answer using two significant figures. ANSWER: = 2710 = 0.987 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 6 of 16 3/23/2015 11:12 AM Correct Part B Find the cat’s velocity at = 8.0 . Express your answer using two significant figures. ANSWER: Correct Part C What is the cat’s acceleration at ? Express your answer using two significant figures. ANSWER: Correct Part D What is the cat’s acceleration at ? Express your answer using two significant figures. ANSWER: Correct Part E What is the cat’s acceleration at ? Express your answer using two significant figures. ANSWER: = 1.3 = -2.7 = -1.3 = -1.3 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 7 of 16 3/23/2015 11:12 AM Correct Part F What distance does the cat move during the first 4.5 ? Express your answer using two significant figures. ANSWER: Correct Part G What distance does the cat move from to ? Express your answer using two significant figures. ANSWER: Correct Part H Sketch clear graph of the cat’s acceleration as function of time, assuming that the cat started at the origin. ANSWER: = -1.3 = 23 = 26 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 8 of 16 3/23/2015 11:12 AM Correct Part I Sketch clear graph of the cat’s position as function of time, assuming that the cat started at the origin. ANSWER: Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 9 of 16 3/23/2015 11:12 AM All attempts used; correct answer displayed Exercise 2.35 Part A If a flea can jump straight up to a height of 0.510 , what is its initial speed as it leaves the ground? ANSWER: Correct Part B How long is it in the air? ANSWER: Correct = 3.16 = 0.645 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 10 of 16 3/23/2015 11:12 AM Exercise 2.36 A small rock is thrown vertically upward with a speed of 18.0 from the edge of the roof of a 39.0 tall building. The rock doesn’t hit the building on its way back down and lands in the street below. Air resistance can be neglected. Part A What is the speed of the rock just before it hits the street? Express your answer with the appropriate units. ANSWER: Correct Part B How much time elapses from when the rock is thrown until it hits the street? Express your answer with the appropriate units. ANSWER: Correct Exercise 2.38 You throw a glob of putty straight up toward the ceiling, which is 3.00 above the point where the putty leaves your hand. The initial speed of the putty as it leaves your hand is 9.70 . Part A What is the speed of the putty just before it strikes the ceiling? Express your answer with the appropriate units. ANSWER: Correct Part B = 33.0 = 5.20 = 5.94 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 11 of 16 3/23/2015 11:12 AM How much time from when it leaves your hand does it take the putty to reach the ceiling? Express your answer with the appropriate units. ANSWER: Correct Exercise 3.1 A squirrel has x- and y-coordinates ( 1.2 , 3.3 ) at time and coordinates ( 5.3 , -0.80 ) at time = 2.6 . Part A For this time interval, find the x-component of the average velocity. Express your answer using two significant figures. ANSWER: Correct Part B For this time interval, find the y-component of the average velocity. Express your answer using two significant figures. ANSWER: Correct Part C Find the magnitude of the average velocity. Express your answer using two significant figures. ANSWER: = 0.384 = 1.6 = -1.6 = 2.2 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 12 of 16 3/23/2015 11:12 AM Correct Part D Find the direction of the average velocity. Express your answer using two significant figures. ANSWER: Correct Exercise 3.3 A web page designer creates an animation in which a dot on a computer screen has a position of 4.1 2.1 4.7 . Part A Find the average velocity of the dot between and . Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Part B Find the instantaneous velocity at . Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Part C = 45 below the x-axis = 4.2,4.7 = 0,4.7 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 13 of 16 3/23/2015 11:12 AM Find the instantaneous velocity at . Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Part D Find the instantaneous velocity at . Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Exercise 3.5 A jet plane is flying at a constant altitude. At time it has components of velocity 89 , 108 . At time 32.5 the components are 165 , 37 . Part A For this time interval calculate the average acceleration. Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Part B Find the magnitude of the average acceleration. Express your answer using two significant figures. ANSWER: = 4.2,4.7 = 8.4,4.7 = 2.3,-2.2 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 14 of 16 3/23/2015 11:12 AM Correct Part C Find the direction of the average acceleration (let the direction be the angle that the vector makes with the +x-axis, measured counterclockwise). ANSWER: Correct Exercise 3.4 The position of a squirrel running in a park is given by . Part A What is , the -component of the velocity of the squirrel, as function of time? ANSWER: Correct Part B What is , the y-component of the velocity of the squirrel, as function of time? ANSWER: = 3.2 = -43.1 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 15 of 16 3/23/2015 11:12 AM Correct Part C At 4.51 , how far is the squirrel from its initial position? Express your answer to three significant figures and include the appropriate units. ANSWER: All attempts used; correct answer displayed Part D At 4.51 , what is the magnitude of the squirrel’s velocity? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part E At 4.51 , what is the direction (in degrees counterclockwise from +x-axis) of the squirrel’s velocity? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Score Summary: Your score on this assignment is 90.1%. You received 14.42 out of a possible total of 16 points. = 2.65 = 1.31 = 62.5 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 16 of 16 3/23/2015 11:12 AM