## Assignment 2 Conditional Probability, Bayes Theorem, and Random Variables Conditional Probability and Bayes’ Theorem Problems 1-14 from Problem Set on Conditional Probability and Bayes’ Theorem I am including all the question here so that there is no confusion. Q1. Pair of six sided dices are rolled and the outcome is noted: What is the sample space? What is the size of the sample space? Suppose all we are interested in is the sum of the two outcomes. What is the probability that the sum of the two is 6? 7? 8? (Note: This can be solved using both enumeration and conditional probability method). Here, it makes more sense to use the enumeration approach than conditional probability. It is, however, listed here to set the stage for Q5. What is the probability that the sum of the two is above 5 and the two numbers are equal? Express this question in terms of events A, B, and set operators. What is the probability that the sum of the two is above 5 or the two numbers are equal? Express this question in terms of events A, B, and set operators. Q2. If P(A)=0.4, P(B)=0.5 and P(A∩B)=0.3 What is the value of (a) P(A|B) and (b) P(B|A) Q3. At a fair, a vendor has 25 helium balloons on strings: 10 balloons are yellow, 8 are red, and 7 are green. A balloon is selected at random and sold. Given that the balloon sold is yellow, what is the probability that the next balloon selected at random is also yellow? Q4. A bowl contains seven blue chips and three red chips. Two chips are to be drawn at random and without replacement. What is the probability that the fist chip is a red chip and the second a blue? Express this question in terms of events A, B, and set operators and use conditional probability. Q5. Three six sided dices are rolled and the outcome is noted: What is the size of the sample space? What is the probability that the sum of the three numbers is 6? 13? 18? Solve using conditional probability How does the concept of conditional probability help? Q6. A grade school boy has 5 blue and four white marbles in his left pocket and four blue and five white marbles in his right pocket. If he transfers one marble at random from his left pocket to his right pocket, what is the probability of his then drawing a blue marble from his right pocket? Q7. In a certain factory, machine I, II, and III are all producing springs of the same length. Of their production, machines I, II, and III produce 2%, 1%, and 3% defective springs respectively. Of the total production of springs in the factory, machine I produces 35%, machine II produces 25%, and machine III produces 40%. If one spring is selected at random from the total springs produced in a day, what is the probability that it is defective? Given that the selected spring is defective, what is the probability that it was produced on machine III? Q8. Bowl B1 contains 2 white chips, bowl B2 contains 2 red chips, bowl B3 contains 2 white and 2 red chips, and Bowl B4 contains 3 white chips and 1 red chip. The probabilities of selecting bowl B1, B2, B3, and B4 are 1/2, 1/4, 1/8, and 1/8 respectively. A bowl is selected using these probabilities, and a chip is then drawn at random. Find P(W), the probability of drawing a white chip P(B1|W): the probability that bowl B1 was selected, given that a white chip was drawn. Q9. A pap smear is a screening procedure used to detect cervical cancer. For women with this cancer, there are about 16% false negative. For women without cervical cancer, there are about 19% false positive. In the US, there are about 8 women in 100,000 who have this cancer. What is the probability that a woman who has been tested positive actually has cervical cancer? Q10. There is a new diagnostic test for a disease that occurs in about 0.05% of the population. The test is not perfect but will detect a person with the disease 99% of the time. It will, however, say that a person without the disease has the disease about 3% of the time. A person is selected at random from the population and the test indicates that this person has the disease. What are the conditional probabilities that The person has the disease The person does not have the disease Q11. Consider two urns: the first contains two white and seven black balls, and the second contains five white and six black balls. We flip a fair coin and then draw a ball from the first urn or the second urn depending on whether the outcome was a head or a tails. What is the conditional probability that the outcome of the toss was heads given that a white ball was selected? Q12. In answering a question on a multiple-choice test a student either knows the answer or guesses. Let p be the probability that she knows the answer. Assume that a student who guesses at the answer will be correct with probability 1/m where m is the number of multiple choice alternatives. What is the conditional probability that a student knew the answer given that she answered it correctly? Q13. A laboratory blood test is 95% effective in detecting a certain disease when it is, in fact, present. However, the test also yields a “false positive” result for 1% of the healthy persons tested (i.e., if a healthy person is tested, then, with probability 0.01, the test result will imply that he has the disease.). If 0.5% of the population actually have the disease, what is the probability a person has the disease given that his test results are positive? Q14. An urn contains b black balls and r red balls. One of the balls is drawn at random, but when it is put back in the urn, c additional balls of the same color are put in it with it. Now suppose that we draw another ball. What is the probability that the first ball drawn was black given that the second ball drawn was red? Random Variables Q15. Suppose an experiment consists of tossing two six sided fair dice and observing the outcomes. What is the sample space? Let Y denote the sum of the two numbers that appear on the dice. Define Y to be a random variable. What are the values that the random variable Y can take? What does it mean if we say Y=7? What does it mean if I say that Y<7? Q16. Suppose an experiment consists of picking a sample of size n from a population of size N. Assume that n≪N. Also, assume that the population contains D defective parts and N-D non defective parts, where n

## Phys4A: Practice problems for the 1st midterm test Fall 2015 1 If K has dimensions ML2/T 2, the k in K = kmv 2 must be: Answer: dimensionless 2. A 8.7 hour trip is made at an average speed of 73.0 km/h. If the first third of the trip (chronologically) was driven at 96.5 km/h, what was the average speed for the rest of the journey? Answer: 61 km/h 3. A car travels 95 km to the north at 70.0 km/h, then turns around and travels 21.9 km at 80.0 km/h. What is the difference between the average speed and the average velocity on this trip? Answer: 27 km/h 4. A particle confined to motion along the x axis moves with constant acceleration from x = 2.0 m to x = 8.0 m during a 2.5s time interval. The velocity of the particle at x = 8.0 m is 2.8 m/s. What is the acceleration during this time interval? Answer: 0.32 m/s2 5. A package is dropped from a helicopter moving upward at 15 m/s. If it takes 16.0 s before the package strikes the ground, how high above the ground was the package when it was released? (Disregard air resistance.) Answer: 1000m 6. If vector B is added to vector A, the result is 6i + j. If B is subtracted from A, the result is –4i + 7j. What is the magnitude of A? Answer: 4.1 7. Starting from one oasis, a camel walks 25 km in a direction 30° south of west and then walks 30 km toward the north to a second oasis. What is the direction from the first oasis to the second oasis? Answer: 51° W of N 8 A river 1.00 mile wide flows with a constant speed of 1.00 mi/h. A man can row a boat at 2.00 mi/h. He crosses the river in a direction that puts him directly across the river from the starting point, and then he returns in a direction that puts him back at the starting point in the shortest time possible. The travel time for the man is, Answer: 1.15 h 9 An airplane is heading due east. The airspeed indicator shows that the plane is moving at a speed of 370 km/h relative to the air. If the wind is blowing from the south at 92.5 km/h, the velocity of the airplane relative to the ground is: Answer: 381 km/h at 76o east of north 10. A rock is projected from the edge of the top of a building with an initial velocity of 12.2 m/s at an angle of 53° above the horizontal. The rock strikes the ground a horizontal distance of 25 m from the base of the building. Assume that the ground is level and that the side of the building is vertical. How tall is the building? Answer: 23.6m 11. A boy throws a rock with an initial velocity of 3.13 m/s at 30.0° above the horizontal. How long does it take for the rock to reach the maximum height of its trajectory? Answer: 0.160 s 12. A helicopter is traveling at 54 m/s at a constant altitude of 100 m over a level field. If a wheel falls off the helicopter, with what speed will it hit the ground? Note: air resistance negligible. Answer: 70 m/s 13 A rescue airplane is diving at an angle of 37º below the horizontal with a speed of 250 m/s. It releases a survival package when it is at an altitude of 600 m. If air resistance is ignored, the horizontal distance of the point of impact from the plane at the moment of the package’s release is, Answer: 720 m 14. A hobby rocket reaches a height of 72.3 m and lands 111 m from the launch point. What was the angle of launch? Answer: 69.0° 15. An object moving at a constant speed requires 6.0 s to go once around a circle with a diameter of 4.0 m. What is the magnitude of the instantaneous acceleration of the particle during this time? Answer: 2.2 m/s2 16 A ball is whirled in a horizontal circle of radius r and speed v. The radius is increased to 2r keeping the speed of the ball constant. The period of the ball changes by a factor of Answer: two

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## Chapter 9 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, April 18, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Momentum and Internal Forces Learning Goal: To understand the concept of total momentum for a system of objects and the effect of the internal forces on the total momentum. We begin by introducing the following terms: System: Any collection of objects, either pointlike or extended. In many momentum-related problems, you have a certain freedom in choosing the objects to be considered as your system. Making a wise choice is often a crucial step in solving the problem. Internal force: Any force interaction between two objects belonging to the chosen system. Let us stress that both interacting objects must belong to the system. External force: Any force interaction between objects at least one of which does not belong to the chosen system; in other words, at least one of the objects is external to the system. Closed system: a system that is not subject to any external forces. Total momentum: The vector sum of the individual momenta of all objects constituting the system. In this problem, you will analyze a system composed of two blocks, 1 and 2, of respective masses and . To simplify the analysis, we will make several assumptions: The blocks can move in only one dimension, namely, 1. along the x axis. 2. The masses of the blocks remain constant. 3. The system is closed. At time , the x components of the velocity and the acceleration of block 1 are denoted by and . Similarly, the x components of the velocity and acceleration of block 2 are denoted by and . In this problem, you will show that the total momentum of the system is not changed by the presence of internal forces. m1 m2 t v1(t) a1 (t) v2 (t) a2 (t) Part A Find , the x component of the total momentum of the system at time . Express your answer in terms of , , , and . ANSWER: Part B Find the time derivative of the x component of the system’s total momentum. Express your answer in terms of , , , and . You did not open hints for this part. ANSWER: Why did we bother with all this math? The expression for the derivative of momentum that we just obtained will be useful in reaching our desired conclusion, if only for this very special case. Part C The quantity (mass times acceleration) is dimensionally equivalent to which of the following? ANSWER: p(t) t m1 m2 v1 (t) v2 (t) p(t) = dp(t)/dt a1 (t) a2 (t) m1 m2 dp(t)/dt = ma Part D Acceleration is due to which of the following physical quantities? ANSWER: Part E Since we have assumed that the system composed of blocks 1 and 2 is closed, what could be the reason for the acceleration of block 1? You did not open hints for this part. ANSWER: momentum energy force acceleration inertia velocity speed energy momentum force Part F This question will be shown after you complete previous question(s). Part G Let us denote the x component of the force exerted by block 1 on block 2 by , and the x component of the force exerted by block 2 on block 1 by . Which of the following pairs equalities is a direct consequence of Newton’s second law? ANSWER: Part H Let us recall that we have denoted the force exerted by block 1 on block 2 by , and the force exerted by block 2 on block 1 by . If we suppose that is greater than , which of the following statements about forces is true? You did not open hints for this part. the large mass of block 1 air resistance Earth’s gravitational attraction a force exerted by block 2 on block 1 a force exerted by block 1 on block 2 F12 F21 and and and and F12 = m2a2 F21 = m1a1 F12 = m1a1 F21 = m2a2 F12 = m1a2 F21 = m2a1 F12 = m2a1 F21 = m1a2 F12 F21 m1 m2 ANSWER: Part I Now recall the expression for the time derivative of the x component of the system’s total momentum: . Considering the information that you now have, choose the best alternative for an equivalent expression to . You did not open hints for this part. ANSWER: Impulse and Momentum Ranking Task Six automobiles are initially traveling at the indicated velocities. The automobiles have different masses and velocities. The drivers step on the brakes and all automobiles are brought to rest. Part A Rank these automobiles based on the magnitude of their momentum before the brakes are applied, from largest to smallest. Rank from largest to smallest. To rank items as equivalent, overlap them. If the ranking cannot be determined, check the box below. ANSWER: Both forces have equal magnitudes. |F12 | > |F21| |F21 | > |F12| dpx(t)/dt = Fx dpx(t)/dt 0 nonzero constant kt kt2 Part B Rank these automobiles based on the magnitude of the impulse needed to stop them, from largest to smallest. Rank from largest to smallest. To rank items as equivalent, overlap them. If the ranking cannot be determined, check the box below. You did not open hints for this part. ANSWER: Part C Rank the automobiles based on the magnitude of the force needed to stop them, from largest to smallest. Rank from largest to smallest. To rank items as equivalent, overlap them. If the ranking cannot be determined, check the box below. You did not open hints for this part. ANSWER: A Game of Frictionless Catch Chuck and Jackie stand on separate carts, both of which can slide without friction. The combined mass of Chuck and his cart, , is identical to the combined mass of Jackie and her cart. Initially, Chuck and Jackie and their carts are at rest. Chuck then picks up a ball of mass and throws it to Jackie, who catches it. Assume that the ball travels in a straight line parallel to the ground (ignore the effect of gravity). After Chuck throws the ball, his speed relative to the ground is . The speed of the thrown ball relative to the ground is . Jackie catches the ball when it reaches her, and she and her cart begin to move. Jackie’s speed relative to the ground after she catches the ball is . When answering the questions in this problem, keep the following in mind: The original mass of Chuck and his cart does not include the 1. mass of the ball. 2. The speed of an object is the magnitude of its velocity. An object’s speed will always be a nonnegative quantity. mcart mball vc vb vj mcart Part A Find the relative speed between Chuck and the ball after Chuck has thrown the ball. Express the speed in terms of and . You did not open hints for this part. ANSWER: Part B What is the speed of the ball (relative to the ground) while it is in the air? Express your answer in terms of , , and . You did not open hints for this part. ANSWER: Part C What is Chuck’s speed (relative to the ground) after he throws the ball? Express your answer in terms of , , and . u vc vb u = vb mball mcart u vb = vc mball mcart u You did not open hints for this part. ANSWER: Part D Find Jackie’s speed (relative to the ground) after she catches the ball, in terms of . Express in terms of , , and . You did not open hints for this part. ANSWER: Part E Find Jackie’s speed (relative to the ground) after she catches the ball, in terms of . Express in terms of , , and . You did not open hints for this part. ANSWER: vc = vj vb vj mball mcart vb vj = vj u vj mball mcart u Momentum in an Explosion A giant “egg” explodes as part of a fireworks display. The egg is at rest before the explosion, and after the explosion, it breaks into two pieces, with the masses indicated in the diagram, traveling in opposite directions. Part A What is the momentum of piece A before the explosion? Express your answer numerically in kilogram meters per second. You did not open hints for this part. ANSWER: vj = pA,i Part B During the explosion, is the force of piece A on piece B greater than, less than, or equal to the force of piece B on piece A? You did not open hints for this part. ANSWER: Part C The momentum of piece B is measured to be 500 after the explosion. Find the momentum of piece A after the explosion. Enter your answer numerically in kilogram meters per second. You did not open hints for this part. ANSWER: pA,i = kg m/s greater than less than equal to cannot be determined kg m/s pA,f pA,f = kg m/s ± PSS 9.1 Conservation of Momentum Learning Goal: To practice Problem-Solving Strategy 9.1 for conservation of momentum problems. An 80- quarterback jumps straight up in the air right before throwing a 0.43- football horizontally at 15 . How fast will he be moving backward just after releasing the ball? PROBLEM-SOLVING STRATEGY 9.1 Conservation of momentum MODEL: Clearly define the system. If possible, choose a system that is isolated ( ) or within which the interactions are sufficiently short and intense that you can ignore external forces for the duration of the interaction (the impulse approximation). Momentum is conserved. If it is not possible to choose an isolated system, try to divide the problem into parts such that momentum is conserved during one segment of the motion. Other segments of the motion can be analyzed using Newton’s laws or, as you will learn later, conservation of energy. VISUALIZE: Draw a before-and-after pictorial representation. Define symbols that will be used in the problem, list known values, and identify what you are trying to find. SOLVE: The mathematical representation is based on the law of conservation of momentum: . In component form, this is ASSESS: Check that your result has the correct units, is reasonable, and answers the question. Model The interaction at study in this problem is the action of throwing the ball, performed by the quarterback while being off the ground. To apply conservation of momentum to this interaction, you will need to clearly define a system that is isolated or within which the impulse approximation can be applied. Part A Sort the following objects as part of the system or not. Drag the appropriate objects to their respective bins. ANSWER: kg kg m/s F = net 0 P = f P i (pfx + ( + ( += ( + ( + ( + )1 pfx)2 pfx)3 pix)1 pix)2 pix)3 (pfy + ( + ( += ( + ( + ( + )1 pfy)2 pfy)3 piy)1 piy)2 piy)3 Part B This question will be shown after you complete previous question(s). Visualize Solve Part C This question will be shown after you complete previous question(s). Assess Part D This question will be shown after you complete previous question(s). Conservation of Momentum in Inelastic Collisions Learning Goal: To understand the vector nature of momentum in the case in which two objects collide and stick together. In this problem we will consider a collision of two moving objects such that after the collision, the objects stick together and travel off as a single unit. The collision is therefore completely inelastic. You have probably learned that “momentum is conserved” in an inelastic collision. But how does this fact help you to solve collision problems? The following questions should help you to clarify the meaning and implications of the statement “momentum is conserved.” Part A What physical quantities are conserved in this collision? ANSWER: Part B Two cars of equal mass collide inelastically and stick together after the collision. Before the collision, their speeds are and . What is the speed of the two-car system after the collision? the magnitude of the momentum only the net momentum (considered as a vector) only the momentum of each object considered individually v1 v2 You did not open hints for this part. ANSWER: Part C Two cars collide inelastically and stick together after the collision. Before the collision, the magnitudes of their momenta are and . After the collision, what is the magnitude of their combined momentum? You did not open hints for this part. ANSWER: The answer depends on the directions in which the cars were moving before the collision. v1 + v2 v1 − v2 v2 − v1 v1v2 −−−− ” v1+v2 2 v1 + 2 v2 2 −−−−−−− p1 p2 Part D Two cars collide inelastically and stick together after the collision. Before the collision, their momenta are and . After the collision, their combined momentum is . Of what can one be certain? You did not open hints for this part. ANSWER: Part E Two cars collide inelastically and stick together after the collision. Before the collision, the magnitudes of their momenta are and . After the collision, the magnitude of their combined momentum is . Of what can one be certain? The answer depends on the directions in which the cars were moving before the collision. p1 + p2 p1 − p2 p2 − p1 p1p2 −−−− ” p1+p2 2 p1 + 2 p2 2 −−−−−−− p 1 p 2 p p = p1 + # p2 # p = p1 − # p2 # p = p2 − # p1 # p1 p2 p You did not open hints for this part. ANSWER: Colliding Cars In this problem we will consider the collision of two cars initially moving at right angles. We assume that after the collision the cars stick together and travel off as a single unit. The collision is therefore completely inelastic. Two cars of masses and collide at an intersection. Before the collision, car 1 was traveling eastward at a speed of , and car 2 was traveling northward at a speed of . After the collision, the two cars stick together and travel off in the direction shown. Part A p1 + p2 $ p $ p1p2 −−−− ” p1 +p2 $ p $ p1+p2 2 p1 + p2 $ p $ |p1 − p2 | p1 + p2 $ p $ p1 + 2 p2 2 −−−−−−− m1 m2 v1 v2 First, find the magnitude of , that is, the speed of the two-car unit after the collision. Express in terms of , , and the cars’ initial speeds and . You did not open hints for this part. ANSWER: Part B Find the tangent of the angle . Express your answer in terms of the momenta of the two cars, and . ANSWER: Part C Suppose that after the collision, ; in other words, is . This means that before the collision: ANSWER: v v v m1 m2 v1 v2 v = p1 p2 tan( ) = tan = 1 45′ The magnitudes of the momenta of the cars were equal. The masses of the cars were equal. The velocities of the cars were equal. ± Catching a Ball on Ice Olaf is standing on a sheet of ice that covers the football stadium parking lot in Buffalo, New York; there is negligible friction between his feet and the ice. A friend throws Olaf a ball of mass 0.400 that is traveling horizontally at 11.2 . Olaf’s mass is 67.1 . Part A If Olaf catches the ball, with what speed do Olaf and the ball move afterward? Express your answer numerically in meters per second. You did not open hints for this part. ANSWER: Part B kg m/s kg vf vf = m/s If the ball hits Olaf and bounces off his chest horizontally at 8.00 in the opposite direction, what is his speed after the collision? Express your answer numerically in meters per second. You did not open hints for this part. ANSWER: A One-Dimensional Inelastic Collision Block 1, of mass = 2.90 , moves along a frictionless air track with speed = 25.0 . It collides with block 2, of mass = 17.0 , which was initially at rest. The blocks stick together after the collision. Part A Find the magnitude of the total initial momentum of the two-block system. Express your answer numerically. m/s vf vf = m/s m1 kg v1 m/s m2 kg pi You did not open hints for this part. ANSWER: Part B Find , the magnitude of the final velocity of the two-block system. Express your answer numerically. You did not open hints for this part. ANSWER: Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. pi = kg m/s vf vf = m/s

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## Assignment 3 Due: 11:59pm on Friday, February 14, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Problem 2.68 As a science project, you drop a watermelon off the top of the Empire State Building, 320 m above the sidewalk. It so happens that Superman flies by at the instant you release the watermelon. Superman is headed straight down with a speed of 36.0 . Part A How fast is the watermelon going when it passes Superman? Express your answer with the appropriate units. ANSWER: Correct Problem 2.63 A motorist is driving at when she sees that a traffic light ahead has just turned red. She knows that this light stays red for , and she wants to reach the light just as it turns green again. It takes her to step on the brakes and begin slowing. Part A What is her speed as she reaches the light at the instant it turns green? Express your answer with the appropriate units. ANSWER: m/s 72.0 ms 20 m/s 200 m 15 s 1.0 s 5.71 ms Correct Conceptual Question 4.1 Part A At this instant, is the particle in the figurespeeding up, slowing down, or traveling at constant speed? ANSWER: Correct Part B Is this particle curving to the right, curving to the left, or traveling straight? Speeding up Slowing down Traveling at constant speed ANSWER: Correct Conceptual Question 4.2 Part A At this instant, is the particle in the following figure speeding up, slowing down, or traveling at constant speed? ANSWER: Curving to the right Curving to the left Traveling straight Correct Part B Is this particle curving upward, curving downward, or traveling straight? ANSWER: Correct Problem 4.8 A particle’s trajectory is described by and , where is in s. Part A What is the particle’s speed at ? ANSWER: The particle is speeding up. The particle is slowing down. The particle is traveling at constant speed. The particle is curving upward. The particle is curving downward. The particle is traveling straight. x = ( 1 −2 ) m 2 t3 t2 y = ( 1 −2t) m 2 t2 t t = 0 s v = 2 m/s Correct Part B What is the particle’s speed at ? Express your answer using two significant figures. ANSWER: Correct Part C What is the particle’s direction of motion, measured as an angle from the x-axis, at ? Express your answer using two significant figures. ANSWER: Correct Part D What is the particle’s direction of motion, measured as an angle from the x-axis, at ? Express your answer using two significant figures. ANSWER: t = 5.0s v = 18 m/s t = 0 s = -90 counterclockwise from the +x axis. t = 5.0s = 9.7 counterclockwise from the +x axis. Correct Problem 4.9 A rocket-powered hockey puck moves on a horizontal frictionless table. The figure shows the graph of and the figure shows the graph of , the x- and y-components of the puck’s velocity, respectively. The puck starts at the origin. Part A In which direction is the puck moving at = 3 ? Give your answer as an angle from the x-axis. Express your answer using two significant figures. ANSWER: Correct Part B vx vy t s = 51 above the x-axis How far from the origin is the puck at 5 ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Enhanced EOC: Problem 4.13 A rifle is aimed horizontally at a target 51.0 away. The bullet hits the target 1.50 below the aim point. You may want to review ( pages 91 – 95) . For help with math skills, you may want to review: Quadratic Equations Part A What was the bullet’s flight time? Express your answer with the appropriate units. Hint 1. How to approach the problem Start by drawing a picture of the bullet’s trajectory, including where it leaves the gun and where it hits the target. You can assume that the gun was held parallel to the ground. Label the distances given in the problem. Choose an x-y coordinate system, making sure to label the origin. It is conventional to have x in the horizontal direction and y in the vertical direction. What is the y coordinate when the bullet leaves the gun? What is the y coordinate when it hits the target? What is the initial velocity in the y direction? What is the acceleration in the y direction? What is the equation that describes the motion in the vertical y direction as a function of time? Can you use the equation for to determine the time of flight? Why was it not necessary to include the motion in the x direction? s s = 180 cm m cm y(t) y(t) ANSWER: Correct Part B What was the bullet’s speed as it left the barrel? Express your answer with the appropriate units. Hint 1. How to approach the problem In the coordinate system introduced in Part A, what are the x coordinates when the bullet leaves the gun and when it hits the target? Is there any acceleration in the x direction? What is the equation that describes the motion in the horizontal x direction as a function of time? Can you use the equation for to determine the initial velocity? ANSWER: Correct Introduction to Projectile Motion Learning Goal: To understand the basic concepts of projectile motion. Projectile motion may seem rather complex at first. However, by breaking it down into components, you will find that it is really no different than the one-dimensional motions that you have already studied. One of the most often used techniques in physics is to divide two- and three-dimensional quantities into components. For instance, in projectile motion, a particle has some initial velocity . In general, this velocity can point in any direction on the xy plane and can have any magnitude. To make a problem more managable, it is common to break up such a quantity into its x component and its y component . 5.53×10−2 s x(t) x(t) 922 ms v vx vy Consider a particle with initial velocity that has magnitude 12.0 and is directed 60.0 above the negative x axis. Part A What is the x component of ? Express your answer in meters per second. ANSWER: Correct Part B What is the y component of ? Express your answer in meters per second. ANSWER: Correct Breaking up the velocities into components is particularly useful when the components do not affect each other. Eventually, you will learn about situations in which the components of velocity do affect one another, but for now you will only be looking at problems where they do not. So, if there is acceleration in the x direction but not in the y direction, then the x component of the velocity will change, but the y component of the velocity will not. Part C Look at this applet. The motion diagram for a projectile is displayed, as are the motion diagrams for each component. The x-component motion diagram is what you would get if you shined a spotlight down on the particle as it moved and recorded the motion of its shadow. Similarly, if you shined a spotlight to the left and recorded the particle’s shadow, you would get the motion diagram for its y component. How would you describe the two motion diagrams for the components? ANSWER: v m/s degrees vx v vx = -6.00 m/s vy v vy = 10.4 m/s Correct As you can see, the two components of the motion obey their own independent kinematic laws. For the vertical component, there is an acceleration downward with magnitude . Thus, you can calculate the vertical position of the particle at any time using the standard kinematic equation . Similarly, there is no acceleration in the horizontal direction, so the horizontal position of the particle is given by the standard kinematic equation . Now, consider this applet. Two balls are simultaneously dropped from a height of 5.0 . Part D How long does it take for the balls to reach the ground? Use 10 for the magnitude of the acceleration due to gravity. Express your answer in seconds to two significant figures. Hint 1. How to approach the problem The balls are released from rest at a height of 5.0 at time . Using these numbers and basic kinematics, you can determine the amount of time it takes for the balls to reach the ground. ANSWER: Correct This situation, which you have dealt with before (motion under the constant acceleration of gravity), is actually a special case of projectile motion. Think of this as projectile motion where the horizontal component of the initial velocity is zero. Both the vertical and horizontal components exhibit motion with constant nonzero acceleration. The vertical component exhibits motion with constant nonzero acceleration, whereas the horizontal component exhibits constant-velocity motion. The vertical component exhibits constant-velocity motion, whereas the horizontal component exhibits motion with constant nonzero acceleration. Both the vertical and horizontal components exhibit motion with constant velocity. g = 10 m/s2 y = y0 + v0 t + (1/2)at2 x = x0 + v0 t m tg m/s2 m t = 0 s tg = 1.0 s Part E Imagine the ball on the left is given a nonzero initial speed in the horizontal direction, while the ball on the right continues to fall with zero initial velocity. What horizontal speed must the ball on the left start with so that it hits the ground at the same position as the ball on the right? Remember that the two balls are released, starting a horizontal distance of 3.0 apart. Express your answer in meters per second to two significant figures. Hint 1. How to approach the problem Recall from Part B that the horizontal component of velocity does not change during projectile motion. Therefore, you need to find the horizontal component of velocity such that, in a time , the ball will move horizontally 3.0 . You can assume that its initial x coordinate is . ANSWER: Correct You can adjust the horizontal speeds in this applet. Notice that regardless of what horizontal speeds you give to the balls, they continue to move vertically in the same way (i.e., they are at the same y coordinate at the same time). Problem 4.12 A ball thrown horizontally at 27 travels a horizontal distance of 49 before hitting the ground. Part A From what height was the ball thrown? Express your answer using two significant figures with the appropriate units. ANSWER: vx m vx tg = 1.0 s m x0 = 0.0 m vx = 3.0 m/s m/s m h = 16 m Correct Enhanced EOC: Problem 4.20 The figure shows the angular-velocity-versus-time graph for a particle moving in a circle. You may want to review ( page ) . For help with math skills, you may want to review: The Definite Integral Part A How many revolutions does the object make during the first 3.5 ? Express your answer using two significant figures. You did not open hints for this part. ANSWER: s n = Incorrect; Try Again Problem 4.26 To withstand “g-forces” of up to 10 g’s, caused by suddenly pulling out of a steep dive, fighter jet pilots train on a “human centrifuge.” 10 g’s is an acceleration of . Part A If the length of the centrifuge arm is 10.0 , at what speed is the rider moving when she experiences 10 g’s? Express your answer with the appropriate units. ANSWER: Correct Problem 4.28 Your roommate is working on his bicycle and has the bike upside down. He spins the 60.0 -diameter wheel, and you notice that a pebble stuck in the tread goes by three times every second. Part A What is the pebble’s speed? Express your answer with the appropriate units. ANSWER: Correct 98 m/s2 m 31.3 ms cm 5.65 ms Part B What is the pebble’s acceleration? Express your answer with the appropriate units. ANSWER: Correct Enhanced EOC: Problem 4.43 On the Apollo 14 mission to the moon, astronaut Alan Shepard hit a golf ball with a 6 iron. The acceleration due to gravity on the moon is 1/6 of its value on earth. Suppose he hits the ball with a speed of 13 at an angle 50 above the horizontal. You may want to review ( pages 90 – 95) . For help with math skills, you may want to review: Quadratic Equations Part A How much farther did the ball travel on the moon than it would have on earth? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Start by drawing a picture of the path of the golf ball, showing its starting and ending points. Choose a coordinate system, and label the origin. It is conventional to let x be the horizontal direction and y the vertical direction. What is the initial velocity in the x and y directions? What is the acceleration in the x and y directions on the moon and on the earth? What are the equations for and as a function of time, and , respectively? What is the y coordinate when the golf ball hits the ground? Can you use this information to determine the time of flight on the moon and on the earth? 107 m s2 m/s x y x(t) y(t) Once you have the time of flight, how can you use the equation to determine the total distance traveled? Compare the distance traveled on the moon to the distance traveled on the earth . ANSWER: Correct Part B For how much more time was the ball in flight? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the equation describing as a function of time? What is the initial x component of the ball’s velocity? How are the initial x component of the ball’s velocity and the distance traveled related to the time of flight? What is the difference between the time of flight on the moon and on earth? ANSWER: Correct Problem 4.42 In the Olympic shotput event, an athlete throws the shot with an initial speed of 12 at a 40.0 angle from the horizontal. The shot leaves her hand at a height of 1.8 above the ground. x(t) L = 85 m x(t) x t = 10 s m/s m Part A How far does the shot travel? Express your answer to four significant figures and include the appropriate units. ANSWER: Correct Part B Repeat the calculation of part (a) for angles of 42.5 , 45.0 , and 47.5 . Express your answer to four significant figures and include the appropriate units. ANSWER: Correct Part C Express your answer to four significant figures and include the appropriate units. ANSWER: Correct Part D x = 16.36 m x(42.5 ) = 16.39 m x(45.0 ) = 16.31 m Express your answer to four significant figures and include the appropriate units. ANSWER: Correct Part E At what angle of release does she throw the farthest? ANSWER: Correct Problem 4.44 A ball is thrown toward a cliff of height with a speed of 32 and an angle of 60 above horizontal. It lands on the edge of the cliff 3.2 later. Part A How high is the cliff? Express your answer to two significant figures and include the appropriate units. ANSWER: x(47.5 ) = 16.13 m 40.0 42.5 45.0 47.5 h m/s s h = 39 m Answer Requested Part B What was the maximum height of the ball? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the ball’s impact speed? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 4.58 A typical laboratory centrifuge rotates at 3600 . Test tubes have to be placed into a centrifuge very carefully because of the very large accelerations. Part A What is the acceleration at the end of a test tube that is 10 from the axis of rotation? Express your answer with the appropriate units. hmax = 39 m v = 16 ms rpm cm ANSWER: Correct Part B For comparison, what is the magnitude of the acceleration a test tube would experience if dropped from a height of 1.0 and stopped in a 1.7-ms-long encounter with a hard floor? Express your answer with the appropriate units. ANSWER: Correct Problem 4.62 Communications satellites are placed in a circular orbit where they stay directly over a fixed point on the equator as the earth rotates. These are called geosynchronous orbits. The radius of the earth is , and the altitude of a geosynchronous orbit is ( 22000 miles). Part A What is the speed of a satellite in a geosynchronous orbit? Express your answer with the appropriate units. ANSWER: Correct a = 1.42×104 m s2 m a = 2610 m s2 6.37 × 106m 3.58 × 107m v = 3070 ms Part B What is the magnitude of the acceleration of a satellite in a geosynchronous orbit? Express your answer with the appropriate units. ANSWER: Correct Score Summary: Your score on this assignment is 89.5%. You received 103.82 out of a possible total of 116 points. a = 0.223 m s2

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## Assignment 7 Due: 11:59pm on Friday, March 21, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 8.5 The figure shows two balls of equal mass moving in vertical circles. Part A Is the tension in string A greater than, less than, or equal to the tension in string B if the balls travel over the top of the circle with equal speed? ANSWER: Correct The tension in string A is less than the tension in string B. The tension in string A is equal to the tension in string B. The tension in string A is greater than the tension in string B. Part B Is the tension in string A greater than, less than, or equal to the tension in string B if the balls travel over the top of the circle with equal angular velocity? ANSWER: Correct A Mass on a Turntable: Conceptual A small metal cylinder rests on a circular turntable that is rotating at a constant rate, as illustrated in the diagram. Part A Which of the following sets of vectors best describes the velocity, acceleration, and net force acting on the cylinder at the point indicated in the diagram? The tension in string A is less than the tension in string B. The tension in string A is equal to the tension in string B. The tension in string A is greater than the tension in string B. Typesetting math: 100% Hint 1. The direction of acceleration can be determined from Newton’s second law According to Newton’s second law, the acceleration of an object has the same direction as the net force acting on that object. ANSWER: Correct Part B Let be the distance between the cylinder and the center of the turntable. Now assume that the cylinder is moved to a new location from the center of the turntable. Which of the following statements accurately describe the motion of the cylinder at the new location? Check all that apply. a b c d e R R/2 Typesetting math: 100% Hint 1. Find the speed of the cylinder Find the speed of the cylinder at the new location. Assume that the cylinder makes one complete turn in a period of time . Express your answer in terms of and . ANSWER: Hint 2. Find the acceleration of the cylinder Find the magnitude of the acceleration of the cylinder at the new location. Assume that the cylinder makes one complete turn in a period of time . Express your answer in terms of and . Hint 1. Centripetal acceleration Recall that the acceleration of an object that moves in a circular path of radius with constant speed has magnitude given by . Note that both the velocity and radius of the trajectory change when the cylinder is moved. ANSWER: ANSWER: v T R T v = R T a T R T r v a = v2 r a = 22R T 2 Typesetting math: 100% Correct Accelerating along a Racetrack A road race is taking place along the track shown in the figure . All of the cars are moving at constant speeds. The car at point F is traveling along a straight section of the track, whereas all the other cars are moving along curved segments of the track. Part A Let be the velocity of the car at point A. What can you say about the acceleration of the car at that point? Hint 1. Acceleration along a curved path The speed of the cylinder has decreased. The speed of the cylinder has increased. The magnitude of the acceleration of the cylinder has decreased. The magnitude of the acceleration of the cylinder has increased. The speed and the acceleration of the cylinder have not changed. v A Typesetting math: 100% Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero acceleration because the direction of its velocity is changing, even though the magnitude of its velocity (the speed) is constant. Moreover, if the speed is constant, the object’s acceleration is always perpendicular to the velocity vector at each point along the curved path and is directed toward the center of curvature of the path. ANSWER: Correct Part B Let be the velocity of the car at point C. What can you say about the acceleration of the car at that point? Hint 1. Acceleration along a curved path Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero acceleration because the direction of its velocity is changing, even though the magnitude of its velocity (the speed) is constant. Moreover, if the speed is constant, the object’s acceleration is always perpendicular to the velocity vector at each point along the curved path and is directed toward the center of curvature of the path. ANSWER: v v The acceleration is parallel to . The acceleration is perpendicular to and directed toward the inside of the track. The acceleration is perpendicular to and directed toward the outside of the track. The acceleration is neither parallel nor perpendicular to . The acceleration is zero. v A v A v A v A v C v v Typesetting math: 100% Correct Part C Let be the velocity of the car at point D. What can you say about the acceleration of the car at that point? Hint 1. Acceleration along a curved path Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero acceleration because the direction of its velocity is changing, even though the magnitude of its velocity (the speed) is constant. Moreover, if the speed is constant, the object’s acceleration is always perpendicular to the velocity vector at each point along the curved path and is directed toward the center of curvature of the path. ANSWER: Correct The acceleration is parallel to . The acceleration is perpendicular to and pointed toward the inside of the track. The acceleration is perpendicular to and pointed toward the outside of the track. The acceleration is neither parallel nor perpendicular to . The acceleration is zero. v C v C v C v C v D v v The acceleration is parallel to . The acceleration is perpendicular to and pointed toward the inside of the track. The acceleration is perpendicular to and pointed toward the outside of the track. The acceleration is neither parallel nor perpendicular to . The acceleration is zero. v D v D v D v D Typesetting math: 100% Part D Let be the velocity of the car at point F. What can you say about the acceleration of the car at that point? Hint 1. Acceleration along a straight path The velocity of an object that moves along a straight path is always parallel to the direction of the path, and the object has a nonzero acceleration only if the magnitude of its velocity changes in time. ANSWER: Correct Part E Assuming that all cars have equal speeds, which car has the acceleration of the greatest magnitude, and which one has the acceleration of the least magnitude? Use A for the car at point A, B for the car at point B, and so on. Express your answer as the name the car that has the greatest magnitude of acceleration followed by the car with the least magnitude of accelation, and separate your answers with a comma. Hint 1. How to approach the problem Recall that the magnitude of the acceleration of an object that moves at constant speed along a curved path is inversely proportional to the radius of curvature of the path. ANSWER: v F The acceleration is parallel to . The acceleration is perpendicular to and pointed toward the inside of the track. The acceleration is perpendicular to and pointed toward the outside of the track. The acceleration is neither parallel nor perpendicular to . The acceleration is zero. v F v F v F v F Typesetting math: 100% Correct Part F Assume that the car at point A and the one at point E are traveling along circular paths that have the same radius. If the car at point A now moves twice as fast as the car at point E, how is the magnitude of its acceleration related to that of car E. Hint 1. Find the acceleration of the car at point E Let be the radius of the two curves along which the cars at points A and E are traveling. What is the magnitude of the acceleration of the car at point E? Express your answer in terms of the radius of curvature and the speed of car E. Hint 1. Uniform circular motion The magnitude of the acceleration of an object that moves with constant speed along a circular path of radius is given by . ANSWER: Hint 2. Find the acceleration of the car at point A If , what is the acceleration of the car at point A? Let be the radius of the two curves along which the cars at points A and E are traveling. Express your answer in terms of the speed of the car at E and the radius . r aE r vE a v r a = v2 r aE = vE 2 r vA = 2vE aA r vE r Typesetting math: 100% Hint 1. Uniform circular motion The magnitude of the acceleration of an object that moves with constant speed along a circular path of radius is given by . ANSWER: ANSWER: Correct Problem 8.5 A 1300 car takes a 50- -radius unbanked curve at 13 . Part A What is the size of the friction force on the car? Express your answer to two significant figures and include the appropriate units. ANSWER: v r a = v2 r aA = 4vE 2 r The magnitude of the acceleration of the car at point A is twice that of the car at point E. The magnitude of the acceleration of the car at point A is the same as that of the car at point E. The magnitude of the acceleration of the car at point A is half that of the car at point E. The magnitude of the acceleration of the car at point A is four times that of the car at point E. kg m m/s Typesetting math: 100% Correct Problem 8.10 It is proposed that future space stations create an artificial gravity by rotating. Suppose a space station is constructed as a 1600- -diameter cylinder that rotates about its axis. The inside surface is the deck of the space station. Part A What rotation period will provide “normal” gravity? Express your answer with the appropriate units. ANSWER: Correct Problem 8.7 In the Bohr model of the hydrogen atom, an electron orbits a proton at a distance of . The proton pulls on the electron with an electric force of . Part A How many revolutions per second does the electron make? Express your answer with the appropriate units. ANSWER: fs = 4400 N m T = 56.8 s (mass m = 9.1 × 10−31 kg) 5.3 × 10−11 m 8.2 × 10−8 N Typesetting math: 100% Correct Problem 8.14 The weight of passengers on a roller coaster increases by 56 as the car goes through a dip with a 38 radius of curvature. Part A What is the car’s speed at the bottom of the dip? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 8.18 While at the county fair, you decide to ride the Ferris wheel. Having eaten too many candy apples and elephant ears, you find the motion somewhat unpleasant. To take your mind off your stomach, you wonder about the motion of the ride. You estimate the radius of the big wheel to be 14 , and you use your watch to find that each loop around takes 24 . Part A What is your speed? Express your answer to two significant figures and include the appropriate units. ANSWER: 6.56×1015 rev s % m v = 14 ms m s v = 3.7 ms Typesetting math: 100% Correct Part B What is the magnitude of your acceleration? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the ratio of your weight at the top of the ride to your weight while standing on the ground? Express your answer using two significant figures. ANSWER: Correct Part D What is the ratio of your weight at the bottom of the ride to your weight while standing on the ground? Express your answer using two significant figures. ANSWER: a = 0.96 m s2 = 0.90 wtop FG Typesetting math: 100% Correct Enhanced EOC: Problem 8.46 A heavy ball with a weight of 120 is hung from the ceiling of a lecture hall on a 4.4- -long rope. The ball is pulled to one side and released to swing as a pendulum, reaching a speed of 5.6 as it passes through the lowest point. You may want to review ( pages 201 – 204) . For help with math skills, you may want to review: Solutions of Systems of Equations Part A What is the tension in the rope at that point? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Start by drawing a free-body diagram indicating the forces acting on the ball when it is at its lowest point. Choose a coordinate system. What is the direction of the acceleration in your chosen coordinate system? What is the magnitude of the acceleration for the mass, which is moving in a circular path? What is Newton’s second law applied to the mass at the bottom of its swing? Make sure to use your coordinate system when determining the signs of all the forces and the acceleration. What is the tension in the rope at this point? ANSWER: = 1.1 wbottom FG N m m/s T = 210 N Typesetting math: 100% Correct Problem 8.43 In an amusement park ride called The Roundup, passengers stand inside a 16.0 -diameter rotating ring. After the ring has acquired sufficient speed, it tilts into a vertical plane, as shown in the figure . Part A Suppose the ring rotates once every 4.80 . If a rider’s mass is 54.0 , with how much force does the ring push on her at the top of the ride? Express your answer with the appropriate units. ANSWER: Correct Part B m s kg 211 N Typesetting math: 100% Suppose the ring rotates once every 4.80 . If a rider’s mass is 54.0 , with how much force does the ring push on her at the bottom of the ride? Express your answer with the appropriate units. ANSWER: Correct Part C What is the longest rotation period of the wheel that will prevent the riders from falling off at the top? Express your answer with the appropriate units. ANSWER: Correct Conceptual Question 9.9 A 2 object is moving to the right with a speed of 1 when it experiences an impulse of 6 . Part A What is the object’s speed after the impulse? Express your answer as an integer and include the appropriate units. ANSWER: s kg 1270 N 5.68 s kg m/s i ^ Ns i ^ v = 4 ms Typesetting math: 100% Correct Part B What is the object’s direction after the impulse? ANSWER: Correct Conceptual Question 9.10 A 2 object is moving to the right with a speed of 2 when it experiences an impulse of -6 . Part A What is the object’s speed after the impulse? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part B What is the object’s direction after the impulse? to the right to the left kg m/s i ^ Ns i ^ v = 1 ms Typesetting math: 100% ANSWER: Correct Problem 9.5 Part A In the figure , what value of gives an impulse of 6.4 ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct to the right to the left Fmax Ns Fmax = 1.6×103 N Typesetting math: 100% Impulse on a Baseball Learning Goal: To understand the relationship between force, impulse, and momentum. The effect of a net force acting on an object is related both to the force and to the total time the force acts on the object. The physical quantity impulse is a measure of both these effects. For a constant net force, the impulse is given by . The impulse is a vector pointing in the same direction as the force vector. The units of are or . Recall that when a net force acts on an object, the object will accelerate, causing a change in its velocity. Hence the object’s momentum ( ) will also change. The impulse-momentum theorem describes the effect that an impulse has on an object’s motion: . So the change in momentum of an object equals the net impulse, that is, the net force multiplied by the time over which the force acts. A given change in momentum can result from a large force over a short time or a smaller force over a longer time. In Parts A, B, C consider the following situation. In a baseball game the batter swings and gets a good solid hit. His swing applies a force of 12,000 to the ball for a time of . Part A Assuming that this force is constant, what is the magnitude of the impulse on the ball? Enter your answer numerically in newton seconds using two significant figures. ANSWER: Correct We often visualize the impulse by drawing a graph of force versus time. For a constant net force such as that used in the previous part, the graph will look like the one shown in the figure. (F J J = F) t J N * s kg * m/s p = mv )p = J = F) t N 0.70 × 10−3 s J J = 8.4 N * s Typesetting math: 100% Part B The net force versus time graph has a rectangular shape. Often in physics geometric properties of graphs have physical meaning. ANSWER: Correct The assumption of a constant net force is idealized to make the problem easier to solve. A real force, especially in a case like the one presented in Parts A and B, where a large force is applied for a short time, is not likely to be constant. A more realistic graph of the force that the swinging bat applies to the baseball will show the force building up to a maximum value as the bat comes into full contact with the ball. Then as the ball loses contact with the bat, the graph will show the force decaying to zero. It will look like the graph in the figure. For this graph, the length height area slope of the rectangle corresponds to the impulse. Typesetting math: 100% Part C If both the graph representing the constant net force and the graph representing the variable net force represent the same impulse acting on the baseball, which geometric properties must the two graphs have in common? ANSWER: maximum force area slope Typesetting math: 100% Correct When the net force varies over time, as in the case of the real net force acting on the baseball, you can simplify the problem by finding the average net force acting on the baseball during time . This average net force is treated as a constant force that acts on the ball for time . The impulse on the ball can then be found as . Graphically, this method states that the impulse of the baseball can be represented by either the area under the net force versus time curve or the area under the average net force versus time curve. These areas are represented in the figure as the areas shaded in red and blue respectively. The impulse of an object is also related to its change in momentum. Once the impulse is known, it can be used to find the change in momentum, or if either the initial or final momentum is known, the other momentum can be found. Keep in mind that . Because both impulse and momentum are vectors, it is essential to account for the direction of each vector, even in a one-dimensional problem. Part D Assume that a pitcher throws a baseball so that it travels in a straight line parallel to the ground. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. Define the direction the pitcher originally throws the ball as the +x direction. ANSWER: F avg )t )t J = F )t avg J = )p = m(vf − vi ) Typesetting math: 100% Correct Part E Now assume that the pitcher in Part D throws a 0.145- baseball parallel to the ground with a speed of 32 in the +x direction. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. What is the ball’s velocity just after leaving the bat if the bat applies an impulse of to the baseball? Enter your answer numerically in meters per second using two significant figures. ANSWER: Correct The negative sign in the answer indicates that after the bat hits the ball, the ball travels in the opposite direction to that defined to be positive. Problem 9.9 A 2.6 object is moving to the right with a speed of 1.0 when it experiences the force shown in the figure. The impulse on the ball caused by the bat will be in the positive negative x direction. kg m/s −8.4 N * s v = -26 m/s kg m/s Typesetting math: 100% Part A What is the object’s speed after the force ends? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the object’s direction after the force ends? ANSWER: Correct Enhanced EOC: Problem 9.27 A tennis player swings her 1000 g racket with a speed of 11.0 . She hits a 60 g tennis ball that was approaching her at a speed of 19.0 . The ball rebounds at 41.0 . You may want to review ( pages 226 – 232) . For help with math skills, you may want to review: v = 0.62 ms to the right to the left m/s m/s m/s Typesetting math: 100% Solving Algebraic Equations Part A How fast is her racket moving immediately after the impact? You can ignore the interaction of the racket with her hand for the brief duration of the collision. Express your answer with the appropriate units. Hint 1. How to approach the problem Given that you can ignore the interaction of the racket with her hand during the collision, what is conserved during the collision? Draw a picture indicating the direction of the racket and ball before the collision and a separate picture for after the collision. Place a coordinate system on your pictures, indicating the positive x direction. Keeping in mind that velocity can be either positive or negative in your coordinate system, what is the initial momentum of the ball–racket system? What is the final momentum of the ball–racket system in terms of the velocity of the racket after the collision? Using conservation of momentum, what are the velocity and speed of the racket after the collision? ANSWER: Correct Part B If the tennis ball and racket are in contact for 8.00 , what is the average force that the racket exerts on the ball? Express your answer with the appropriate units. Hint 1. How to approach the problem How is the impulse on the ball related to the change in momentum of the ball? What is the change in momentum of the ball? How are the impulse on the ball and the collision time related to the average force on the ball? 7.40 ms ms Typesetting math: 100% ANSWER: Correct Problem 9.14 A 2.00×104 railroad car is rolling at 6.00 when a 6000 load of gravel is suddenly dropped in. Part A What is the car’s speed just after the gravel is loaded? Express your answer with the appropriate units. ANSWER: Correct Problem 9.17 A 330 bird flying along at 5.0 sees a 9.0 insect heading straight toward it with a speed of 34 (as measured by an observer on the ground, not by the bird). The bird opens its mouth wide and enjoys a nice lunch. Part A What is the bird’s speed immediately after swallowing? Express your answer to two significant figures and include the appropriate units. ANSWER: 450 N kg m/s kg 4.62 ms g m/s g m/s Typesetting math: 100% Correct Problem 9.20 A 50.0 archer, standing on frictionless ice, shoots a 200 arrow at a speed of 200 . Part A What is the recoil speed of the archer? Express your answer with the appropriate units. ANSWER: Correct Problem 9.25 A 40.0 ball of clay traveling east at 4.50 collides and sticks together with a 50.0 ball of clay traveling north at 4.50 . Part A What is the speed of the resulting ball of clay? Express your answer with the appropriate units. ANSWER: v = 4.0 ms kg g m/s 0.800 ms g m/s g m/s 3.20 ms Typesetting math: 100% Correct Problem 9.32 A particle of mass is at rest at . Its momentum for is given by , where is in . Part A Find an expression for , the force exerted on the particle as a function of time. Express your answer in terms of the given quantities. ANSWER: Correct Problem 9.37 Most geologists believe that the dinosaurs became extinct 65 million years ago when a large comet or asteroid struck the earth, throwing up so much dust that the sun was blocked out for a period of many months. Suppose an asteroid with a diameter of 2.0 and a mass of 1.2×1013 hits the earth with an impact speed of 4.5×104 . Part A What is the earth’s recoil speed after such a collision? (Use a reference frame in which the earth was initially at rest.) Assume that . Express your answer to two significant figures and include the appropriate units. ANSWER: m t = 0 t > 0 px = 6t2 kgm/s t s Fx(t) Fx = 12t N km kg m/s MEarth= 5.98 × 1024 kg = 9.0×10−8 v ms Typesetting math: 100% Correct Part B What percentage is this of the earth’s speed around the sun? (Use the astronomical data in the textbook.) Express your answer using two significant figures. ANSWER: Correct Problem 9.42 One billiard ball is shot east at 1.8 . A second, identical billiard ball is shot west at 1.2 . The balls have a glancing collision, not a head-on collision, deflecting the second ball by 90 and sending it north at 1.50 . Part A What is the speed of the first ball after the collision? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the direction of the first ball after the collision? Give the direction as an angle south of east. = 3.0×10−10 of v % the earth’s speed m/s m/s 1 m/s v = 1.6 ms Typesetting math: 100% Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 9.49 Two 490 blocks of wood are 2.0 apart on a frictionless table. A 12 bullet is fired at 420 toward the blocks. It passes all the way through the first block, then embeds itself in the second block. The speed of the first block immediately afterward is 5.6 . Part A What is the speed of the second block after the bullet stops? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Score Summary: Your score on this assignment is 99.5%. You received 156.21 out of a possible total of 157 points. = 68 1 g m g m/s m/s v = 4.6 ms Typesetting math: 100%

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## Assignment 6 Due: 11:59pm on Friday, March 7, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 7.7 A small car is pushing a large truck. They are speeding up. Part A Is the force of the truck on the car larger than, smaller than, or equal to the force of the car on the truck? ANSWER: Correct Conceptual Question 7.12 The figure shows two masses at rest. The string is massless and the pulley is frictionless. The spring scale reads in . Assume that = 4 . The force of the truck on the car is larger than the force of the car on the truck. The force of the truck on the car is equal to the force of the car on the truck. The force of the truck on the car is smaller than the force of the car on the truck. kg m kg Part A What is the reading of the scale? Express your answer to one significant figure and include the appropriate units. ANSWER: Correct Problem 7.1 A weightlifter stands up at constant speed from a squatting position while holding a heavy barbell across his shoulders. Part A Draw a free-body diagram for the barbells. 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: m = 4 kg Correct Part B Draw a free-body diagram for the weight lifter. 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: Correct Problem 7.6 Block A in the figure is sliding down the incline. The rope is massless, and the massless pulley turns on frictionless bearings, but the surface is not frictionless. The rope and the pulley are among the interacting objects, but you’ll have to decide if they’re part of the system. Part A Draw a free-body diagram for the block A. The orientation of your vectors will be graded. The exact length of your vectors will not be graded. ANSWER: Correct Part B Draw a free-body diagram for the block B. The orientation of your vectors will be graded. The exact length of your vectors will not be graded. ANSWER: Correct A Space Walk Part A An astronaut is taking a space walk near the shuttle when her safety tether breaks. What should the astronaut do to get back to the shuttle? Hint 1. How to approach the problem Newton’s 3rd law tells us that forces occur in pairs. Within each pair, the forces, often called action and reaction, have equal magnitude and opposite direction. Which of the actions suggested in the problem will result in the force pushing the astronaut back to the shuttle? ANSWER: Correct As the astronaut throws the tool away from the shuttle, she exerts a force in the direction away from the shuttle. Then, by Newton’s 3rd law, the tool will exert an opposite force on her. Thus, as she throws the tool, a force directed toward the shuttle will act on the astronaut. Newton’s 2nd law stipulates that the astronaut would acquire an acceleration toward the shuttle. Part B Assuming that the astronaut can throw any tool with the same force, what tool should be thrown to get back to the shuttle as quickly as possible? You should consider how much mass is left behind as the object is thrown as well as the mass of the object itself. Hint 1. How to approach the problem Recall that the force acting on the astronaut is equal in magnitude and opposite in direction to the force that she exerts on the tool. Hint 2. Newton’s 2nd law Newton’s 2nd law states that . If force is held constant, then acceleration is inversely proportional to mass. For example, when the same force is applied to objects of different mass, the object with the largest mass will experience the smallest acceleration. ANSWER: Attempt to “swim” toward the shuttle. Take slow steps toward the shuttle. Take a tool from her tool belt and throw it away from the shuttle. Take the portion of the safety tether still attached to her belt and throw it toward the shuttle. F = ma Correct The force that acts on the astronaut must equal in magnitude the force that she exerts on the tool. Therefore, if she exerts the same force on any tool, the force acting on her will be independent of the mass of the tool. However, the acceleration that the astronaut would acquire is inversely proportional to her mass since she is acted upon by a constant force. If she throws the tool with the largest mass, the remaining mass (the astronaut plus her remaining tools) would be smallest—and the acceleration the greatest! Part C If the astronaut throws the tool with a force of 16.0 , what is the magnitude of the acceleration of the astronaut during the throw? Assume that the total mass of the astronaut after she throws the tool is 80.0 . Express your answer in meters per second squared. Hint 1. Find the force acting on the astronaut What is the magnitude of the force acting on the astronaut as she throws the tool? Express your answer in newtons. ANSWER: Hint 2. Newton’s 2nd law An object of mass acted upon by a net force has an acceleration given by . ANSWER: The tool with the smallest mass. The tool with the largest mass. Any tool, since the mass of the tool would make no difference. N a kg F F = 16.0 N m F a F = ma a = 0.200 m/s2 Correct Problem 7.10 Blocks with masses of 2 , 4 , and 6 are lined up in a row on a frictionless table. All three are pushed forward by a 60 force applied to the 2 block. Part A How much force does the 4 block exert on the 6 block? Express your answer to one significant figure and include the appropriate units. ANSWER: Correct Part B How much force does the 4 block exert on the 2 block? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 7.9 A 1000 car pushes a 2100 truck that has a dead battery. When the driver steps on the accelerator, the drive wheels of the car push against the ground with a force of 4500 . Rolling friction can kg kg kg N kg kg kg F = 30 N kg kg F = 50 N kg kg N be neglected. Part A What is the magnitude of the force of the car on the truck? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the magnitude of the force of the truck on the car? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Atwood Machine Special Cases An Atwood machine consists of two blocks (of masses and ) tied together with a massless rope that passes over a fixed, perfect (massless and frictionless) pulley. In this problem you’ll investigate some special cases where physical variables describing the Atwood machine take on limiting values. Often, examining special cases will simplify a problem, so that the solution may be found from inspection or from the results of a problem you’ve already seen. For all parts of this problem, take upward to be the positive direction and take the gravitational constant, , to be positive. F = 3000 N F = 3000 N m1 m2 g Part A Consider the case where and are both nonzero, and . Let be the magnitude of the tension in the rope connected to the block of mass , and let be the magnitude of the tension in the rope connected to the block of mass . Which of the following statements is true? ANSWER: Correct Part B Now, consider the special case where the block of mass is not present. Find the magnitude, , of the tension in the rope. Try to do this without equations; instead, think about the physical consequences. Hint 1. How to approach the problem If the block of mass is not present, and the rope connecting the two blocks is massless, will the motion of the block of mass be any different from free fall? Hint 2. Which physical law to use Use Newton’s 2nd law on the block of mass . m1 m2 m2 > m1 T1 m1 T2 m2 is always equal to . is greater than by an amount independent of velocity. is greater than but the difference decreases as the blocks increase in velocity. There is not enough information to determine the relationship between and . T1 T2 T2 T1 T2 T1 T1 T2 m1 T m1 m2 m2 ANSWER: Correct Part C For the same special case (the block of mass not present), what is the acceleration of the block of mass ? Express your answer in terms of , and remember that an upward acceleration should be positive. ANSWER: Correct Part D Next, consider the special case where only the block of mass is present. Find the magnitude, , of the tension in the rope. ANSWER: Correct Part E For the same special case (the block of mass not present) what is the acceleration of the end of the rope where the block of mass would have been attached? Express your answer in terms of , and remember that an upward acceleration should be positive. T = 0 m1 m2 g a2 = -9.80 m1 T T = 0 m2 m2 g ANSWER: Correct Part F Next, consider the special case . What is the magnitude of the tension in the rope connecting the two blocks? Use the variable in your answer instead of or . ANSWER: Correct Part G For the same special case ( ), what is the acceleration of the block of mass ? ANSWER: Correct Part H Finally, suppose , while remains finite. What value does the the magnitude of the tension approach? Hint 1. Acceleration of block of mass a2 = 9.80 m1 = m2 = m m m1 m2 T = mg m1 = m2 = m m2 a2 = 0 m1 m2 m1 As becomes large, the finite tension will have a neglible effect on the acceleration, . If you ignore , you can pretend the rope is gone without changing your results for . As , what value does approach? ANSWER: Hint 2. Acceleration of block of mass As , what value will the acceleration of the block of mass approach? ANSWER: Hint 3. Net force on block of mass What is the magnitude of the net force on the block of mass . Express your answer in terms of , , , and any other given quantities. Take the upward direction to be positive. ANSWER: ANSWER: Correct Imagining what would happen if one or more of the variables approached infinity is often a good way to investigate the behavior of a system. m1 T a1 T a1 m1 a1 a1 = -9.80 m2 m1 m2 a2 = 9.80 m2 Fnet m2 T m2 g Fnet = T − m2g T = 2m2g Problem 7.17 A 5.9 rope hangs from the ceiling. Part A What is the tension at the midpoint of the rope? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 7.23 The sled dog in figure drags sleds A and B across the snow. The coefficient of friction between the sleds and the snow is 0.10. Part A If the tension in rope 1 is 100 , what is the tension in rope 2? Express your answer to two significant figures and include the appropriate units. ANSWER: kg T = 29 N N T2 = 180 N Correct Enhanced EOC: Problem 7.31 Two packages at UPS start sliding down the ramp shown in the figure. Package A has a mass of 4.50 and a coefficient of kinetic friction of 0.200. Package B has a mass of 11.0 and a coefficient of kinetic friction of 0.150. You may want to review ( pages 177 – 181) . For help with math skills, you may want to review: Vector Components Part A How long does it take package A to reach the bottom? Express your answer with the appropriate units. Hint 1. How to approach the problem Start by drawing force identification diagrams for package A and package B separately. What are the four forces acting on each block? Which of the forces are related by Newton’s third law? Draw separate free-body diagrams for block A and for block B. What is a good coordinate system to use to describe the motion of the blocks down the ramp? Label your coordinate system on the free-body diagram. In your coordinate system, compute the x and y components of each force on block A. What are the x and y components of the net force on block A? What are the x and y components of the net force on block B? Given that the coefficient of friction of block A is greater than the coefficient of friction of block B, do you think the blocks will stay together as they slide down the ramp? Assuming that they do stay together, how is the acceleration of the two blocks related? (We can check this assumption later.) Using the components of the forces and Newton’s second law, what is the acceleration of the blocks? What is the initial velocity of the blocks? Given the initial velocity and the acceleration, 20 kg kg how long does it take block A to go the given distance? To check that the blocks do indeed stay together, solve for the force of block B on block A. If the force is directed toward the bottom of the ramp, then the blocks stay together. ANSWER: Correct Problem 7.33 The 1.0 kg block in the figure is tied to the wall with a rope. It sits on top of the 2.0 kg block. The lower block is pulled to the right with a tension force of 20 N. The coefficient of kinetic friction at both the lower and upper surfaces of the 2.0 kg block is = 0.420. 1.48 s μk Part A What is the tension in the rope holding the 1.0 kg block to the wall? Express your answer with the appropriate units. ANSWER: Correct Part B What is the acceleration of the 2.0 kg block? Express your answer with the appropriate units. ANSWER: Correct Problem 7.38 The 100 kg block in figure takes 5.60 to reach the floor after being released from rest. 4.12 N 1.77 m s2 s Part A What is the mass of the block on the left? Express your answer with the appropriate units. ANSWER: Correct Problem 7.41 Figure shows a block of mass m resting on a 20 slope. The block has coefficients of friction 0.82 and 0.51 with the surface. It is connected via a massless string over a massless, frictionless pulley to a hanging block of mass 2.0 . Part A What is the minimum mass that will stick and not slip? 98.7 kg kg m Express your answer to three significant figures and include the appropriate units. ANSWER: Correct If you need to use the rounded answer you submitted here in a subsequent part, instead use the full precision answer and only round as a final step before submitting an answer. Part B If this minimum mass is nudged ever so slightly, it will start being pulled up the incline. What acceleration will it have? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Problem 7.46 A house painter uses the chair and pulley arrangement of the figure to lift himself up the side of a house. The painter’s mass is 75 and the chair’s mass is 12 . m = 1.80 kg a = 1.35 m s2 kg kg Part A With what force must he pull down on the rope in order to accelerate upward at 0.22 ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Score Summary: Your score on this assignment is 98.6%. You received 104.5 out of a possible total of 106 points. m/s2 F = 440 N

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## Chapter 10 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, April 18, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy A One-Dimensional Inelastic Collision Block 1, of mass = 3.70 , moves along a frictionless air track with speed = 15.0 . It collides with block 2, of mass = 19.0 , which was initially at rest. The blocks stick together after the collision. Part A Find the magnitude of the total initial momentum of the two-block system. Express your answer numerically. You did not open hints for this part. ANSWER: m1 kg v1 m/s m2 kg pi Part B Find , the magnitude of the final velocity of the two-block system. Express your answer numerically. You did not open hints for this part. ANSWER: Part C What is the change in the two-block system’s kinetic energy due to the collision? Express your answer numerically in joules. You did not open hints for this part. ANSWER: pi = kg m/s vf vf = m/s K = Kfinal − Kinitial K = J Conservation of Energy Ranking Task Six pendulums of various masses are released from various heights above a tabletop, as shown in the figures below. All the pendulums have the same length and are mounted such that at the vertical position their lowest points are the height of the tabletop and just do not strike the tabletop when released. Assume that the size of each bob is negligible. Part A Rank each pendulum on the basis of its initial gravitational potential energy (before being released) relative to the tabletop. Rank from largest to smallest To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: m h Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Momentum and Kinetic Energy Consider two objects (Object 1 and Object 2) moving in the same direction on a frictionless surface. Object 1 moves with speed and has mass . Object 2 moves with speed and has mass . Part A Which object has the larger magnitude of its momentum? You did not open hints for this part. ANSWER: Part B Which object has the larger kinetic energy? You did not open hints for this part. ANSWER: v1 = v m1 = 2m v2 = 2v m2 = m Object 1 has the greater magnitude of its momentum. Object 2 has the greater magnitude of its momentum. Both objects have the same magnitude of their momenta. Object 1 has the greater kinetic energy. Object 2 has the greater kinetic energy. The objects have the same kinetic energy. Projectile Motion and Conservation of Energy Ranking Task Part A Six baseball throws are shown below. In each case the baseball is thrown at the same initial speed and from the same height above the ground. Assume that the effects of air resistance are negligible. Rank these throws according to the speed of the baseball the instant before it hits the ground. Rank from largest to smallest. To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: H PSS 10.1 Conservation of Mechanical Energy Learning Goal: To practice Problem-Solving Strategy 10.1 for conservation of mechanical energy problems. Tarzan, in one tree, sights Jane in another tree. He grabs the end of a vine with length 20 that makes an angle of 45 with the vertical, steps off his tree limb, and swings down and then up to Jane’s open arms. When he arrives, his vine makes an angle of 30 with the vertical. Determine whether he gives her a tender embrace or knocks her off her limb by calculating Tarzan’s speed just before he reaches Jane. You can ignore air resistance and the mass of the vine. PROBLEM-SOLVING STRATEGY 10.1 Conservation of mechanical energy MODEL: Choose a system without friction or other losses of mechanical energy. m VISUALIZE: Draw a before-and-after pictorial representation. Define symbols that will be used in the problem, list known values, and identify what you’re trying to find. SOLVE: The mathematical representation is based on the law of conservation of mechanical energy: . ASSESS: Check that your result has the correct units, is reasonable, and answers the question. Model The problem does not involve friction, nor are there losses of mechanical energy, so conservation of mechanical energy applies. Model Tarzan and the vine as a pendulum. Visualize Part A Which of the following sketches can be used in drawing a before-and-after pictorial representation? ANSWER: Kf + Uf = Ki + Ui Solve Part B What is Tarzan’s speed just before he reaches Jane? Express your answer in meters per second to two significant figures. You did not open hints for this part. ANSWER: Assess Part C This question will be shown after you complete previous question(s). Bungee Jumping Diagram A Diagram B Diagram C Diagram D vf vf = m/s Kate, a bungee jumper, wants to jump off the edge of a bridge that spans a river below. Kate has a mass , and the surface of the bridge is a height above the water. The bungee cord, which has length when unstretched, will first straighten and then stretch as Kate falls. Assume the following: The bungee cord behaves as an ideal spring once it begins to stretch, with spring constant . Kate doesn’t actually jump but simply steps off the edge of the bridge and falls straight downward. Kate’s height is negligible compared to the length of the bungee cord. Hence, she can be treated as a point particle. Use for the magnitude of the acceleration due to gravity. Part A How far below the bridge will Kate eventually be hanging, once she stops oscillating and comes finally to rest? Assume that she doesn’t touch the water. Express the distance in terms of quantities given in the problem introduction. You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Spinning Mass on a Spring An object of mass is attached to a spring with spring constant whose unstretched length is , and whose far end is fixed to a shaft that is rotating with angular speed . Neglect gravity and assume that the mass rotates with angular speed as shown. When solving this problem use an inertial coordinate system, as drawn here. m h L k g d = M k L Part A Given the angular speed , find the radius at which the mass rotates without moving toward or away from the origin. Express the radius in terms of , , , and . You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C R( ) k L M R( ) = This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). ± Baby Bounce with a Hooke One of the pioneers of modern science, Sir Robert Hooke (1635-1703), studied the elastic properties of springs and formulated the law that bears his name. Hooke found the relationship among the force a spring exerts, , the distance from equilibrium the end of the spring is displaced, , and a number called the spring constant (or, sometimes, the force constant of the spring). According to Hooke, the force of the spring is directly proportional to its displacement from equilibrium, or . In its scalar form, this equation is simply . The negative sign indicates that the force that the spring exerts and its displacement have opposite directions. The value of depends on the geometry and the material of the spring; it can be easily determined experimentally using this scalar equation. Toy makers have always been interested in springs for the entertainment value of the motion they produce. One well-known application is a baby bouncer,which consists of a harness seat for a toddler, attached to a spring. The entire contraption hooks onto the top of a doorway. The idea is for the baby to hang in the seat with his or her feet just touching the ground so that a good push up will get the baby bouncing, providing potentially hours of entertainment. F x k F = −kx F = −kx k Part A The following chart and accompanying graph depict an experiment to determine the spring constant for a baby bouncer. Displacement from equilibrium, ( ) Force exerted on the spring, ( ) 0 0 0.005 2.5 0.010 5.0 0.015 7.5 0.020 10 What is the spring constant of the spring being tested for the baby bouncer? Express your answer to two significant figures in newtons per meter. You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Shooting a ball into a box Two children are trying to shoot a marble of mass into a small box using a spring-loaded gun that is fixed on a table and shoots horizontally from the edge of the table. The edge of the table is a height above the top of the box (the height of which is negligibly small), and the center of the box is a distance from the edge of the table. x m F N k k = N/m m H d The spring has a spring constant . The first child compresses the spring a distance and finds that the marble falls short of its target by a horizontal distance . Part A By what distance, , should the second child compress the spring so that the marble lands in the middle of the box? (Assume that height of the box is negligible, so that there is no chance that the marble will hit the side of the box before it lands in the bottom.) Express the distance in terms of , , , , and . You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). k x1 d12 x2 m k g H d x2 = Elastic Collision in One Dimension Block 1, of mass , moves across a frictionless surface with speed . It collides elastically with block 2, of mass , which is at rest ( ). After the collision, block 1 moves with speed , while block 2 moves with speed . Assume that , so that after the collision, the two objects move off in the direction of the first object before the collision. Part A This collision is elastic. What quantities, if any, are conserved in this collision? You did not open hints for this part. ANSWER: Part B What is the final speed of block 1? m1 ui m2 vi = 0 uf vf m1 > m2 kinetic energy only momentum only kinetic energy and momentum uf Express in terms of , , and . You did not open hints for this part. ANSWER: Part C What is the final speed of block 2? Express in terms of , , and . You did not open hints for this part. ANSWER: Ballistic Pendulum In a ballistic pendulum an object of mass is fired with an initial speed at a pendulum bob. The bob has a mass , which is suspended by a rod of length and negligible mass. After the collision, the pendulum and object stick together and swing to a maximum angular displacement as shown . uf m1 m2 ui uf = vf vf m1 m2 ui vf = m v0 M L Part A Find an expression for , the initial speed of the fired object. Express your answer in terms of some or all of the variables , , , and and the acceleration due to gravity, . You did not open hints for this part. ANSWER: Part B An experiment is done to compare the initial speed of bullets fired from different handguns: a 9.0 and a .44 caliber. The guns are fired into a 10- pendulum bob of length . Assume that the 9.0- bullet has a mass of 6.0 and the .44-caliber bullet has a mass of 12 . If the 9.0- bullet causes the pendulum to swing to a maximum angular displacement of 4.3 and the .44-caliber bullet causes a displacement of 10.1 , find the ratio of the initial speed of the 9.0- bullet to the speed of the .44-caliber bullet, . Express your answer numerically. You did not open hints for this part. ANSWER: v0 m M L g v0 = mm kg L mm g g mm mm (v /( 0 )9.0 v0)44 Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. (v0 )9.0/(v0 )44 =

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## Chapter 4 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, February 14, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Advice for the Quarterback A quarterback is set up to throw the football to a receiver who is running with a constant velocity directly away from the quarterback and is now a distance away from the quarterback. The quarterback figures that the ball must be thrown at an angle to the horizontal and he estimates that the receiver must catch the ball a time interval after it is thrown to avoid having opposition players prevent the receiver from making the catch. In the following you may assume that the ball is thrown and caught at the same height above the level playing field. Assume that the y coordinate of the ball at the instant it is thrown or caught is and that the horizontal position of the quaterback is . Use for the magnitude of the acceleration due to gravity, and use the pictured inertial coordinate system when solving the problem. Part A Find , the vertical component of the velocity of the ball when the quarterback releases it. Express in terms of and . Hint 1. Equation of motion in y direction What is the expression for , the height of the ball as a function of time? Answer in terms of , , and . v r D tc y = 0 x = 0 g v0y v0y tc g y(t) t g v0y ANSWER: Incorrect; Try Again Hint 2. Height at which the ball is caught, Remember that after time the ball was caught at the same height as it had been released. That is, . ANSWER: Answer Requested Part B Find , the initial horizontal component of velocity of the ball. Express your answer for in terms of , , and . Hint 1. Receiver’s position Find , the receiver’s position before he catches the ball. Answer in terms of , , and . ANSWER: Football’s position y(t) = v0yt− g 1 2 t2 y(tc) tc y(tc) = y0 = 0 v0y = gtc 2 v0x v0x D tc vr xr D vr tc xr = D + vrtc Typesetting math: 100% Find , the horizontal distance that the ball travels before reaching the receiver. Answer in terms of and . ANSWER: ANSWER: Answer Requested Part C Find the speed with which the quarterback must throw the ball. Answer in terms of , , , and . Hint 1. How to approach the problem Remember that velocity is a vector; from solving Parts A and B you have the two components, from which you can find the magnitude of this vector. ANSWER: Answer Requested Part D xc v0x tc xc = v0xtc v0x = + D tc vr v0 D tc vr g v0 = ( + ) + D tc vr 2 ( ) gtc 2 2 −−−−−−−−−−−−−−−−−−− Typesetting math: 100% Assuming that the quarterback throws the ball with speed , find the angle above the horizontal at which he should throw it. Your solution should contain an inverse trig function (entered as asin, acos, or atan). Give your answer in terms of already known quantities, , , and . Hint 1. Find angle from and Think of velocity as a vector with Cartesian coordinates and . Find the angle that this vector would make with the x axis using the results of Parts A and B. ANSWER: Answer Requested Direction of Velocity at Various Times in Flight for Projectile Motion Conceptual Question For each of the motions described below, determine the algebraic sign (positive, negative, or zero) of the x component and y component of velocity of the object at the time specified. For all of the motions, the positive x axis points to the right and the positive y axis points upward. Alex, a mountaineer, must leap across a wide crevasse. The other side of the crevasse is below the point from which he leaps, as shown in the figure. Alex leaps horizontally and successfully makes the jump. v0 v0x v0y v0 v0x v0y v0xx^ v0yy^ = atan( ) v0y v0x Typesetting math: 100% Part A Determine the algebraic sign of Alex’s x velocity and y velocity at the instant he leaves the ground at the beginning of the jump. Type the algebraic signs of the x velocity and the y velocity separated by a comma (examples: +,- and 0,+). Typesetting math: 100% Hint 1. Algebraic sign of velocity The algebraic sign of the velocity is determined solely by comparing the direction in which the object is moving with the direction that is defined to be positive. In this example, to the right is defined to be the positive x direction and upward the positive y direction. Therefore, any object moving to the right, whether speeding up, slowing down, or even simultaneously moving upward or downward, has a positive x velocity. Similarly, if the object is moving downward, regardless of any other aspect of its motion, its y velocity is negative. Hint 2. Sketch Alex’s initial velocity On the diagram below, sketch the vector representing Alex’s velocity the instant after he leaves the ground at the beginning of the jump. ANSWER: ANSWER: Typesetting math: 100% Answer Requested Part B Determine the algebraic signs of Alex’s x velocity and y velocity the instant before he lands at the end of the jump. Type the algebraic signs of the x velocity and the y velocity separated by a comma (examples: +,- and 0,+). Typesetting math: 100% Hint 1. Sketch Alex’s final velocity On the diagram below, sketch the vector representing Alex’s velocity the instant before he safely lands on the other side of the crevasse. ANSWER: Answer Requested ANSWER: Answer Requested Typesetting math: 100% At the buzzer, a basketball player shoots a desperation shot. The ball goes in! Part C Determine the algebraic signs of the ball’s x velocity and y velocity the instant after it leaves the player’s hands. Type the algebraic signs of the x velocity and the y velocity separated by a comma (examples: +,- and 0,+). Hint 1. Sketch the basketball’s initial velocity On the diagram below, sketch the vector representing the velocity of the basketball the instant after it leaves the player’s hands. ANSWER: Typesetting math: 100% ANSWER: Correct Part D Determine the algebraic signs of the ball’s x velocity and y velocity at the ball’s maximum height. Type the algebraic signs of the x velocity and the y velocity separated by a comma (examples: +,- and 0,+). Hint 1. Sketch the basketball’s velocity at maximum height Typesetting math: 100% On the diagram below, sketch the vector representing the velocity of the basketball the instant it reaches its maximum height. ANSWER: ANSWER: Answer Requested PSS 4.1 Projectile Motion Problems Learning Goal: Typesetting math: 100% To practice Problem-Solving Strategy 4.1 for projectile motion problems. A rock thrown with speed 9.00 and launch angle 30.0 (above the horizontal) travels a horizontal distance of = 17.0 before hitting the ground. From what height was the rock thrown? Use the value = 9.810 for the free-fall acceleration. PROBLEM-SOLVING STRATEGY 4.1 Projectile motion problems MODEL: Make simplifying assumptions, such as treating the object as a particle. Is it reasonable to ignore air resistance? VISUALIZE: Use a pictorial representation. Establish a coordinate system with the x axis horizontal and the y axis vertical. Show important points in the motion on a sketch. Define symbols, and identify what you are trying to find. SOLVE: The acceleration is known: and . Thus, the problem becomes one of two-dimensional kinematics. The kinematic equations are , . is the same for the horizontal and vertical components of the motion. Find from one component, and then use that value for the other component. ASSESS: Check that your result has the correct units, is reasonable, and answers the question. Model Start by making simplifying assumptions: Model the rock as a particle in free fall. You can ignore air resistance because the rock is a relatively heavy object moving relatively slowly. Visualize Part A Which diagram represents an accurate sketch of the rock’s trajectory? Hint 1. The launch angle In a projectile’s motion, the angle of the initial velocity above the horizontal is called the launch angle. ANSWER: m/s d m g m/s2 ax = 0 ay = −g xf = xi +vixt, yf = yi +viyt− g(t 1 2 )2 vfx = vix = constant, and vfy = viy − gt t t v i Typesetting math: 100% Typesetting math: 100% Correct Part B As stated in the strategy, choose a coordinate system where the x axis is horizontal and the y axis is vertical. Note that in the strategy, the y component of the projectile’s acceleration, , is taken to be negative. This implies that the positive y axis is upward. Use the same convention for your y axis, and take the positive x axis to be to the right. Where you choose your origin doesn’t change the answer to the question, but choosing an origin can make a problem easier to solve (even if only a bit). Usually it is nice if the majority of the quantities you are given and the quantity you are trying to solve for take positive values relative to your chosen origin. Given this goal, what location for the origin of the coordinate system would make this problem easiest? ANSWER: ay At ground level below the point where the rock is launched At the point where the rock strikes the ground At the peak of the trajectory At the point where the rock is released At ground level below the peak of the trajectory Typesetting math: 100% Correct It’s best to place the origin of the coordinate system at ground level below the launching point because in this way all the points of interest (the launching point and the landing point) will have positive coordinates. (Based on your experience, you know that it’s generally easier to work with positive coordinates.) Keep in mind, however, that this is an arbitrary choice. The correct solution of the problem will not depend on the location of the origin of your coordinate system. Now, define symbols representing initial and final position, velocity, and time. Your target variable is , the initial y coordinate of the rock. Your pictorial representation should be complete now, and similar to the picture below: Solve Part C Find the height from which the rock was launched. Express your answer in meters to three significant figures. yi yi Typesetting math: 100% Hint 1. How to approach the problem The time needed to move horizontally to the final position = 17.0 is the same time needed for the rock to rise from the initial position to the peak of its trajectory and then fall to the ground. Use the information you have about motion in the horizontal direction to solve for . Knowing this time will allow you to use the equations of motion for the vertical direction to solve for . Hint 2. Find the time spent in the air How long ( ) is the rock in the air? Express your answer in seconds to three significant figures. Hint 1. Determine which equation to use Which of the equations given in the strategy and shown below is the most appropriate to calculate the time the rock spent in the air? ANSWER: Hint 2. Find the x component of the initial velocity What is the x component of the rock’s initial velocity? Express your answer in meters per second to three significant figures. ANSWER: ANSWER: t xf = d m yi t yi t t xf = xi + vixt yf = yi + viyt− g(t 1 2 )2 vfy = viy − gt vix = 7.79 m/s Typesetting math: 100% Hint 3. Find the y component of the initial velocity What is the y component of the rock’s initial velocity? Express your answer in meters per second to three significant figures. ANSWER: ANSWER: Answer Requested Assess Part D A second rock is thrown straight upward with a speed 4.500 . If this rock takes 2.181 to fall to the ground, from what height was it released? Express your answer in meters to three significant figures. Hint 1. Identify the known variables What are the values of , , , and for the second rock? Take the positive y axis to be upward and the origin to be located on the ground where the rock lands. Express your answers to four significant figures in the units shown to the right, separated by commas. ANSWER: t = 2.18 s viy = 4.50 m/s yi = 13.5 m m/s s H yf viy t a Typesetting math: 100% Answer Requested Hint 2. Determine which equation to use to find the height Which equation should you use to find ? Keep in mind that if the positive y axis is upward and the origin is located on the ground, . ANSWER: ANSWER: Answer Requested Projectile motion is made up of two independent motions: uniform motion at constant velocity in the horizontal direction and free-fall motion in the vertical direction. Because both rocks were thrown with the same initial vertical velocity, 4.500 , and fell the same vertical distance of 13.5 , they were in the air for the same amount of time. This result was expected and helps to confirm that you did the calculation in Part C correctly. ± Arrow Hits Apple An arrow is shot at an angle of above the horizontal. The arrow hits a tree a horizontal distance away, at the same height above the ground as it was shot. Use for the magnitude of the acceleration due to gravity. Part A , , , = 0,4.500,2.181,-yf viy t a 9.810 m, m/s, s, m/s2 H yi = H yf = yi + viyt− g(t 1 2 )2 vfy = viy − gt = − 2g( − ) v2f y v2i y yf yi H = 13.5 m viy = m/s m = 45 D = 220 m g = 9.8 m/s2 Typesetting math: 100% Find , the time that the arrow spends in the air. Answer numerically in seconds, to two significant figures. Hint 1. Find the initial upward component of velocity in terms of D. Introduce the (unknown) variables and for the initial components of velocity. Then use kinematics to relate them and solve for . What is the vertical component of the initial velocity? Express your answer symbolically in terms of and . Hint 1. Find Find the horizontal component of the initial velocity. Express your answer symbolically in terms of and given symbolic quantities. ANSWER: Hint 2. Find What is the vertical component of the initial velocity? Express your answer symbolically in terms of . ANSWER: ANSWER: ta vy0 vx0 ta vy0 ta D vx0 vx0 ta vx0 = D ta vy0 vy0 vx0 vy0 = vx0 vy0 = D ta Typesetting math: 100% Hint 2. Find the time of flight in terms of the initial vertical component of velocity. From the change in the vertical component of velocity, you should be able to find in terms of and . Give your answer in terms of and . Hint 1. Find When applied to the y-component of velocity, in this problem the formula for with constant acceleration is What is , the vertical component of velocity when the arrow hits the tree? Answer symbolically in terms of only. ANSWER: ANSWER: Hint 3. Put the algebra together to find symbolically. If you have an expression for the initial vertical velocity component in terms in terms of and , and another in terms of and , you should be able to eliminate this initial component to find an expression for Express your answer symbolically in terms of given variables. ANSWER: ta vy0 g vy0 g vy(ta) v(t) −g vy(t) = vy0 − g t vy(ta ) vy0 vy(ta) = −vy0 ta = 2vy0 g ta D ta g ta ta2 t2 = a 2D g Typesetting math: 100% ANSWER: Answer Requested Suppose someone drops an apple from a vertical distance of 6.0 meters, directly above the point where the arrow hits the tree. Part B How long after the arrow was shot should the apple be dropped, in order for the arrow to pierce the apple as the arrow hits the tree? Express your answer numerically in seconds, to two significant figures. Hint 1. When should the apple be dropped The apple should be dropped at the time equal to the total time it takes the arrow to reach the tree minus the time it takes the apple to fall 6.0 meters. Hint 2. Find the time it takes for the apple to fall 6.0 meters How long does it take an apple to fall 6.0 meters? Express your answer numerically in seconds, to two significant figures. ANSWER: Answer Requested ANSWER: ta = 6.7 s tf = 1.1 s td = 5.6 s Typesetting math: 100% Answer Requested Video Tutor: Ball Fired Upward from Accelerating Cart First, launch the video below. You will be asked to use your knowledge of physics to predict the outcome of an experiment. Then, close the video window and answer the questions at right. You can watch the video again at any point. Part A Consider the video you just watched. Suppose we replace the original launcher with one that fires the ball upward at twice the speed. We make no other changes. How far behind the cart will the ball land, compared to the distance in the original experiment? Hint 1. Determine how long the ball is in the air How will doubling the initial upward speed of the ball change the time the ball spends in the air? A kinematic equation may be helpful here. The time in the air will ANSWER: be cut in half. stay the same. double. quadruple. Typesetting math: 100% Hint 2. Determine the appropriate kinematic expression Which of the following kinematic equations correctly describes the horizontal distance between the ball and the cart at the moment the ball lands? The cart’s initial horizontal velocity is , its horizontal acceleration is , and is the time elapsed between launch and impact. ANSWER: ANSWER: Correct The ball will spend twice as much time in the air ( , where is the ball’s initial upward velocity), so it will land four times farther behind the cart: (where is the cart’s horizontal acceleration). Video Tutor: Ball Fired Upward from Moving Cart First, launch the video below. You will be asked to use your knowledge of physics to predict the outcome of an experiment. Then, close the video window and answer the questions at right. You can watch the video again at any point. d v0x ax t d = v0x t d = 1 2 axv0x t2 d = v0x t+ 1 2 axt2 d = 1 2 axt2 the same distance twice as far half as far four times as far by a factor not listed above t = 2v0y/g v0y d = 1 2 axt2 ax Typesetting math: 100% Part A The crew of a cargo plane wishes to drop a crate of supplies on a target below. To hit the target, when should the crew drop the crate? Ignore air resistance. Hint 1. How to approach the problem While the crate is on the plane, it shares the plane’s velocity. What is the crate’s velocity immediately after it is released? Hint 2. What affects the motion of the crate? Gravity will accelerate the crate downward. What, if anything, affects the crate’s horizontal motion? (Keep in mind that we are told to ignore air resistance, even though that’s not very realistic in this situation.) ANSWER: Correct At the moment it is released, the crate shares the plane’s horizontal velocity. In the absence of air resistance, the crate would remain directly below the plane as it fell. Score Summary: Your score on this assignment is 0%. Before the plane is directly over the target After the plane has flown over the target When the plane is directly over the target Typesetting math: 100% You received 0 out of a possible total of 0 points. Typesetting math: 100%

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## 1-Two notions serve as the basis for all torts: wrongs and compensation. True False 2-The goal of tort law is to put a defendant in the position that he or she would have been in had the tort occurred to the defendant. True False 3-Hayley is injured in an accident precipitated by Isolde. Hayley files a tort action against Isolde, seeking to recover for the damage suffered. Damages that are intended to compensate or reimburse a plaintiff for actual losses are: compensatory damages. reimbursement damages. actual damages. punitive damages. 4-Ladd throws a rock intending to hit Minh but misses and hits Nasir instead. On the basis of the tort of battery, Nasir can sue: Ladd. Minh. the rightful owner of the rock. no one. 4-Luella trespasses on Merchandise Mart’s property. Through the use of reasonable force, Merchandise Mart’s security guard detains Luella until the police arrive. Merchandise Mart is liable for: assault. battery. false imprisonment. none of the choice 6-The extreme risk of an activity is a defense against imposing strict liability. True False 7-Misrepresentation in an ad is enough to show an intent to induce the reliance of anyone who may use the product. True False 8-Luke is playing a video game on a defective disk that melts in his game player, starting a fire that injures his hands. Luke files a suit against Mystic Maze, Inc., the game’s maker under the doctrine of strict liability. A significant application of this doctrine is in the area of: cyber torts. intentional torts. product liability. unintentional torts 9-More than two hundred years ago, the Declaration of Independence recognized the importance of protecting creative works. True False 10-n 2014, Cloud Computing Corporation registers its trademark as provided by federal law. After the first renewal, this registration: is renewable every ten years. is renewable every twenty years. runs for life of the corporation plus seventy years. runs forever. 11-Wendy works as a weather announcer for a TV station under the character name Weather Wendy. Wendy can register her character’s name as: a certification mark. a trade name. a service mark. none of the choices 12-Much of the material on the Internet, including software and database information, is not copyrighted. True False 13-In a criminal case, the state must prove its case by a preponderance of the evidence. True False 14-Under the Fourth Amendmentt, general searches through a person’s belongings are permissible. True False 15-Maura enters a gas station and points a gun at the clerk Nate. She then forces Nate to open the cash register and give her all the money. Maura can be charged with: burglary. robbery. larceny. receiving stolen property. 16-Reno, driving while intoxicated, causes a car accident that results in the death of Santo. Reno is arrested and charged with a felony. A felony is a crime punishable by death or imprisonment for: any period of time. more than one year. more than six months. more than ten days. 17-Corporate officers and directors may be held criminally liable for the actions of employees under their supervision. True False 18-Sal assures Tom that she will deliver a truckload of hay to his cattle ranch. A person’s declaration to do a certain act is part of the definition of: an expectation. a moral obligation. a prediction. a promise. 19-Lark promises to buy Mac’s used textbook for $60. Lark is: an offeror. an offeree a promisee. a promisor. 20-Casey offers to sell a certain used forklift to DIY Lumber Outlet, but Casey dies before DIY accepts. Most likely, Casey’s death: did not affect the offer. shortened the time of the offer but did not terminated it. extended the time of the offer. terminated the offer.

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## Name: Date: Quiz IV Vignette 1. Johnny has just come in from recess and he is thirsty. He asks to go to the water fountain, but his teacher tells him that first he has to complete his math worksheet, and then he can have a drink of water. 1. Given that Johnny is thirsty, do you think he will be motivated to complete his math worksheet? 2. Write the correct notation of the 4 term contingency used in this example. Define which piece from the example matches each part of the contingency. 3. What is the MO – and what kind of MO is this? 4. If reinforcement is used in this example – is it positive or negative? Vignette 2. You feel a headache coming on – you see the bottle of advil in your desk drawer. You take the advil. The headache goes away. 5. Write and define the 4 term contingency. 6. What is the MO – and what kind of MO is this? 7. If reinforcement is used in this example – is it positive or negative? —- 8. Define positive reinforcement and give an example. 9. Define negative reinforcement and give an example. Vignette 3. Every time Johnny is given a math worksheet to complete, he kicks, hits, and spits on the teacher. This typically results in Johnny being sent to the principal’s office. 10. How would you label and define this target behavior? 11. What is the probable function of this behavior? 12. What adaptive alternative would you consider teaching Johnny to replace this target behavior? Vignette 4. When Bobby is denied access (told he cannot have) to a preferred toy, he throws himself on the ground, begins screaming and hitting the floor with his fists. This behavioral episode can go on anywhere from 5 to 20 minutes. 13. How would you label and define this target behavior? 14. What is the probable function of this behavior? 15. What type of data collection would you use for this target behavior? — Vignette 5. Johnny knows that when his grandmother watches him, she will try to soothe him with delicious treats if he begins tantrumming. However, he has learned that his mother does NOT give him tasty treats if he engages in problem behavior. Using the 3 term contingency – describe this situation when Grandma is present. (Hint: Does his grandmother function as an SD or an S∆ for tantrumming behavior?) Using the 3 term contingency – describe this situation when his mother is present. (Hint: does his mother function as an SD or an S∆ for tantrum behavior?)

Name: Date: Quiz IV Vignette 1. Johnny … Read More...