• What happens in the business environment when some people function according to those rules, and others do not?

• What happens in the business environment when some people function according to those rules, and others do not?

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5 { GRAVITATION Last Updated: July 16, 2012 Problem List 5.1 Total mass of a shell 5.2 Tunnel through the moon 5.3 Gravitational eld above the center of a thin hoop 5.4 Gravitational force near a metal-cored planet surrounded by a gaseous cloud 5.5 Sphere with linearly increasing mass density 5.6 Jumping o Vesta 5.7 Gravitational force between two massive rods 5.8 Potential energy { Check your answer! 5.9 Ways of solving gravitational problems 5.10 Rod with linearly increasing mass density 5.11 Sphere with constant internal gravitational eld 5.12 Throwing a rock o the moon These problems are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Un- ported License. Please share and/or modify. Back to Problem List 1 5 { GRAVITATION Last Updated: July 16, 2012 5.1 Total mass of a shell Given: Marino { Fall 2011 Consider a spherical shell that extends from r = R to r = 2R with a non-uniform density (r) = 0r. What is the total mass of the shell? Back to Problem List 2 5 { GRAVITATION Last Updated: July 16, 2012 5.2 Tunnel through the moon Given: Marino { Fall 2011 Imagine that NASA digs a straight tunnel through the center of the moon (see gure) to access the Moon’s 3He deposits. An astronaut places a rock in the tunnel at the surface of the moon, and releases it (from rest). Show that the rock obeys the force law for a mass connected to a spring. What is the spring constant? Find the oscillation period for this motion if you assume that Moon has a mass of 7.351022 kg and a radius of 1.74106 m. Assume the moon’s density is uniform throughout its volume, and ignore the moon’s rotation. Given: Pollock { Spring 2011 Imagine (in a parallel universe of unlimited budgets) that NASA digs a straight tunnel through the center of the moon (see gure). A robot place a rock in the tunnel at position r = r0 from the center of the moon, and releases it (from rest). Use Newton’s second law to write the equation of motion of the rock and solve for r(t). Explain in words the rock’s motion. Does the rock return to its initial position at any later time? If so, how long does it takes to return to it? (Give a formula, and a number.) Assume the moon’s density is uniform throughout its volume, and ignore the moon’s rotation. Given: Pollock { Spring 2012 Now lets consider our (real) planet Earth, with total mass M and radius R which we will approximate as a uniform mass density, (r) = 0. (a) Neglecting rotational and frictional e ects, show that a particle dropped into a hole drilled straight through the center of the earth all the way to the far side will oscillate between the two endpoints. (Hint: you will need to set up, and solve, an ODE for the motion) (b) Find the period of the oscillation of this motion. Get a number (in minutes) as a nal result, using data for the earth’s size and mass. (How does that compare to ying to Perth and back?!) Extra Credit: OK, even with unlimited budgets, digging a tunnel through the center of the earth is preposterous. But, suppose instead that the tunnel is a straight-line \chord” through the earth, say directly from New York to Los Angeles. Show that your nal answer for the time taken does not depend on the location of that chord! This is rather remarkable – look again at the time for a free-fall trip (no energy required, except perhaps to compensate for friction) How long would that trip take? Could this work?! Back to Problem List 3 5 { GRAVITATION Last Updated: July 16, 2012 5.3 Gravitational eld above the center of a thin hoop Given: Pollock { Spring 2011, Spring 2012 Consider a very (in nitesimally!) thin but massive loop, radius R (total mass M), centered around the origin, sitting in the x-y plane. Assume it has a uniform linear mass density  (which has units of kg/m) all around it. (So, it’s like a skinny donut that is mostly hole, centered around the z-axis) (a) What is  in terms of M and R? What is the direction of the gravitational eld generated by this mass distribution at a point in space a distance z above the center of the donut, i.e. at (0; 0; z) Explain your reasoning for the direction carefully, try not to simply \wave your hands.” (The answer is extremely intuitive, but can you justify that it is correct?) (b) Compute the gravitational eld, ~g, at the point (0; 0; z) by directly integrating Newton’s law of gravity, summing over all in nitesimal \chunks” of mass along the loop. (c) Compute the gravitational potential at the point (0; 0; z) by directly integrating ?Gdm=r, sum- ming over all in nitesimal \chunks” dm along the loop. Then, take the z-component of the gradient of this potential to check that you agree with your result from the previous part. (d) In the two separate limits z << R and z >> R, Taylor expand your g- eld (in the z-direction)out only to the rst non-zero term, and convince us that both limits make good physical sense. (e) Can you use Gauss’ law to gure out the gravitational potential at the point (0; 0; z)? (If so, do it and check your previous answers. If not, why not?) Extra credit: If you place a small mass a small distance z away from the center, use your Taylor limit for z << R above to write a simple ODE for the equation of motion. Solve it, and discuss the motion Back to Problem List 4 5 { GRAVITATION Last Updated: July 16, 2012 5.4 Gravitational force near a metal-cored planet surrounded by a gaseous cloud Given: Pollock { Spring 2011 Jupiter is composed of a dense spherical core (of liquid metallic hydrogen!) of radius Rc. It is sur- rounded by a spherical cloud of gaseous hydrogen of radius Rg, where Rg > Rc. Let’s assume that the core is of uniform density c and the gaseous cloud is also of uniform density g. What is the gravitational force on an object of mass m that is located at a radius r from the center of Jupiter? Note that you must consider the cases where the object is inside the core, within the gas layer, and outside of the planet. Back to Problem List 5 5 { GRAVITATION Last Updated: July 16, 2012 5.5 Sphere with linearly increasing mass density Given: Pollock { Spring 2011 A planet of mass M and radius R has a nonuniform density that varies with r, the distance from the center according to  = Ar for 0  r  R. (a) What is the constant A in terms of M and R? Does this density pro le strike you as physically plausible, or is just designed as a mathematical exercise? (Brie y, explain) (b) Determine the gravitational force on a satellite of mass m orbiting this planet. In words, please outline the method you plan to use for your solution. (Use the easiest method you can come up with!) In your calculation, you will need to argue that the magnitude of ~g(r; ; ) depends only on r. Be very explicit about this – how do you know that it doesn’t, in fact, depend on  or ? (c) Determine the gravitational force felt by a rock of mass m inside the planet, located at radius r < R. (If the method you use is di erent than in part b, explain why you switched. If not, just proceed!) Explicitly check your result for this part by considering the limits r ! 0 and r ! R. Back to Problem List 6 5 { GRAVITATION Last Updated: July 16, 2012 5.6 Jumping o Vesta Given: Pollock { Spring 2011 You are stranded on the surface of the asteroid Vesta. If the mass of the asteroid is M and its radius is R, how fast would you have to jump o its surface to be able to escape from its gravitational eld? (Your estimate should be based on parameters that characterize the asteroid, not parameters that describe your jumping ability.) Given your formula, look up the approximate mass and radius of the asteroid Vesta 3 and determine a numerical value of the escape velocity. Could you escape in this way? (Brie y, explain) If so, roughly how big in radius is the maximum the asteroid could be, for you to still escape this way? If not, estimate how much smaller an asteroid you would need, to escape from it in this way? Figure 1: Back to Problem List 7 5 { GRAVITATION Last Updated: July 16, 2012 5.7 Gravitational force between two massive rods Given: Pollock { Spring 2011 Consider two identical uniform rods of length L and mass m lying along the same line and having their closest points separated by a distance d as shown in the gure (a) Calculate the mutual force between these rods, both its direction and magnitude. (b) Now do several checks. First, make sure the units worked out (!) The, nd the magnitude of the force in the limit L ! 0. What do you expect? Brie y, discuss. Lastly, nd the magnitude of the force in the limit d ! 1 ? Again, is it what you expect? Brie y, discuss. Figure 2: Given: Pollock { Spring 2012 Determining the gravitational force between two rods: (a) Consider a thin, uniform rod of mass m and length L (and negligible other dimensions) lying on the x axis (from x=-L to 0), as shown in g 1a. Derive a formula for the gravitational eld \g" at any arbitrary point x to the right of the origin (but still on the x-axis!) due to this rod. (b) Now suppose a second rod of length L and mass m sits on the x axis as shown in g 1b, with the left edge a distance \d" away. Calculate the mutual gravitational force between these rods. (c) Let's do some checks! Show that the units work out in parts a and b. Find the magnitude of the force in part a, in the limit x >> L: What do you expect? Brie y, discuss! Finally, verify that your answer to part b gives what you expect in the limit d >> L. ( Hint: This is a bit harder! You need to consistently expand everything to second order, not just rst, because of some interesting cancellations) Fig 1a Fig 1b L m +x x=0 L x=0 x=d m Fig 1a Fig 1b L m +x x=0 L +x x=0 x=d L m m Back to Problem List 8 5 { GRAVITATION Last Updated: July 16, 2012 5.8 Potential energy { Check your answer! Given: Pollock { Spring 2011 On the last exam, we had a problem with a at ring, uniform mass per unit area of , inner radius of R, outer radius of 2R. A satellite (mass m) sat a distance z above the center of the ring. We asked for the gravitational potential energy, and the answer was U(z) = ?2Gm( p 4R2 + z2 ? p R2 + z2) (1) (a) If you are far from the disk (on the z axis), what do you expect for the formula for U(z)? (Don’t say \0″ – as usual, we want the functional form of U(z) as you move far away. Also, explicitly state what we mean by \far away”. (Please don’t compare something with units to something without units!) (b) Show explicitly that the formula above does indeed give precisely the functional dependence you expect. Back to Problem List 9 5 { GRAVITATION Last Updated: July 16, 2012 5.9 Ways of solving gravitational problems Given: Pollock { Spring 2011, Spring 2012 Infinite cylinder ρ=cr x z (a) Half-infinite line mass, uniform linear mass density, λ x (b) R z  P Figure 3: (a) An in nite cylinder of radius R centered on the z-axis, with non-uniform volume mass density  = cr, where r is the radius in cylindrical coordinates. (b) A half-in nite line of mass on the x-axis extending from x = 0 to x = +1, with uniform linear mass density . There are two general methods we use to solve gravitational problems (i.e. nd ~g given some distribution of mass). (a) Describe these two methods. We claim one of these methods is easiest to solve for ~g of mass distribution (a) above, and the other method is easiest to solve for ~g of the mass distribution (b) above. Which method goes with which mass distribution? Please justify your answer. (b) Find ~g of the mass distribution (a) above for any arbitrary point outside the cylinder. (c) Find the x component of the gravitational acceleration, gx, generated by the mass distribution labeled (b) above, at a point P a given distance z up the positive z-axis (as shown). Back to Problem List 10 5 { GRAVITATION Last Updated: July 16, 2012 5.10 Rod with linearly increasing mass density Given: Pollock { Spring 2012 Consider a very (in nitesimally!) thin but massive rod, length L (total mass M), centered around the origin, sitting along the x-axis. (So the left end is at (-L/2, 0,0) and the right end is at (+L/2,0,0) Assume the mass density  (which has units of kg/m)is not uniform, but instead varies linearly with distance from the origin, (x) = cjxj. (a) What is that constant \c” in terms of M and L? What is the direction of the gravitational eld generated by this mass distribution at a point in space a distance z above the center of the rod, i.e. at (0; 0; z) Explain your reasoning for the direction carefully, try not to simply \wave your hands.” (The answer is extremely intuitive, but can you justify that it is correct?) (b) Compute the gravitational eld, ~g, at the point (0; 0; z) by directly integrating Newton’s law of gravity, summing over all in nitesimal \chunks” of mass along the rod. (c) Compute the gravitational potential at the point (0; 0; z) by directly integrating ?Gdm=r, sum- ming over all in nitesimal \chunks” dm along the rod. Then, take the z-component of the gradient of this potential to check that you agree with your result from the previous part. (d) In the limit of large z what do you expect for the functional form for gravitational potential? (Hint: Don’t just say it goes to zero! It’s a rod of mass M, when you’re far away what does it look like? How does it go to zero?) What does \large z” mean here? Use the binomial (or Taylor) expansion to verify that your formula does indeed give exactly what you expect. (Hint: you cannot Taylor expand in something BIG, you have to Taylor expand in something small.) (e) Can you use Gauss’ law to gure out the gravitational potential at the point (0; 0; z)? (If so, do it and check your previous answers. If not, why not?) Back to Problem List 11 5 { GRAVITATION Last Updated: July 16, 2012 5.11 Sphere with constant internal gravitational eld Given: Pollock { Spring 2012 (a) Imagine a planet of total mass M and radius R which has a nonuniform mass density that varies just with r, the distance from the center. For this (admittedly very unusual!) planet, suppose the gravitational eld strength inside the planet turns out to be independent of the radial distance within the sphere. Find the function describing the mass density  = (r) of this planet. (Your nal answer should be written in terms of the given constants.) (b) Now, determine the gravitational force on a satellite of mass m orbiting this planet at distance r > R. (Use the easiest method you can come up with!) Explain your work in words as well as formulas. For instance, in your calculation, you will need to argue that the magnitude of ~g(r; ; ) depends only on r. Be explicit about this – how do you know that it doesn’t, in fact, depend on  or ? (c) As a nal check, explicitly show that your solutions inside and outside the planet (parts a and b) are consistent when r = R. Please also comment on whether this density pro le strikes you as physically plausible, or is it just designed as a mathematical exercise? Defend your reasoning. Back to Problem List 12 5 { GRAVITATION Last Updated: July 16, 2012 5.12 Throwing a rock o the moon Given: Pollock { Spring 2012 Assuming that asteroids have roughly the same mass density as the moon, make an estimate of the largest asteroid that an astronaut could be standing on, and still have a chance of throwing a small object (with their arms, no machinery!) so that it completely escapes the asteroid’s gravitational eld. (This minimum speed is called \escape velocity”) Is the size you computed typical for asteroids in our solar system? Back to Problem List 13

5 { GRAVITATION Last Updated: July 16, 2012 Problem List 5.1 Total mass of a shell 5.2 Tunnel through the moon 5.3 Gravitational eld above the center of a thin hoop 5.4 Gravitational force near a metal-cored planet surrounded by a gaseous cloud 5.5 Sphere with linearly increasing mass density 5.6 Jumping o Vesta 5.7 Gravitational force between two massive rods 5.8 Potential energy { Check your answer! 5.9 Ways of solving gravitational problems 5.10 Rod with linearly increasing mass density 5.11 Sphere with constant internal gravitational eld 5.12 Throwing a rock o the moon These problems are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Un- ported License. Please share and/or modify. Back to Problem List 1 5 { GRAVITATION Last Updated: July 16, 2012 5.1 Total mass of a shell Given: Marino { Fall 2011 Consider a spherical shell that extends from r = R to r = 2R with a non-uniform density (r) = 0r. What is the total mass of the shell? Back to Problem List 2 5 { GRAVITATION Last Updated: July 16, 2012 5.2 Tunnel through the moon Given: Marino { Fall 2011 Imagine that NASA digs a straight tunnel through the center of the moon (see gure) to access the Moon’s 3He deposits. An astronaut places a rock in the tunnel at the surface of the moon, and releases it (from rest). Show that the rock obeys the force law for a mass connected to a spring. What is the spring constant? Find the oscillation period for this motion if you assume that Moon has a mass of 7.351022 kg and a radius of 1.74106 m. Assume the moon’s density is uniform throughout its volume, and ignore the moon’s rotation. Given: Pollock { Spring 2011 Imagine (in a parallel universe of unlimited budgets) that NASA digs a straight tunnel through the center of the moon (see gure). A robot place a rock in the tunnel at position r = r0 from the center of the moon, and releases it (from rest). Use Newton’s second law to write the equation of motion of the rock and solve for r(t). Explain in words the rock’s motion. Does the rock return to its initial position at any later time? If so, how long does it takes to return to it? (Give a formula, and a number.) Assume the moon’s density is uniform throughout its volume, and ignore the moon’s rotation. Given: Pollock { Spring 2012 Now lets consider our (real) planet Earth, with total mass M and radius R which we will approximate as a uniform mass density, (r) = 0. (a) Neglecting rotational and frictional e ects, show that a particle dropped into a hole drilled straight through the center of the earth all the way to the far side will oscillate between the two endpoints. (Hint: you will need to set up, and solve, an ODE for the motion) (b) Find the period of the oscillation of this motion. Get a number (in minutes) as a nal result, using data for the earth’s size and mass. (How does that compare to ying to Perth and back?!) Extra Credit: OK, even with unlimited budgets, digging a tunnel through the center of the earth is preposterous. But, suppose instead that the tunnel is a straight-line \chord” through the earth, say directly from New York to Los Angeles. Show that your nal answer for the time taken does not depend on the location of that chord! This is rather remarkable – look again at the time for a free-fall trip (no energy required, except perhaps to compensate for friction) How long would that trip take? Could this work?! Back to Problem List 3 5 { GRAVITATION Last Updated: July 16, 2012 5.3 Gravitational eld above the center of a thin hoop Given: Pollock { Spring 2011, Spring 2012 Consider a very (in nitesimally!) thin but massive loop, radius R (total mass M), centered around the origin, sitting in the x-y plane. Assume it has a uniform linear mass density  (which has units of kg/m) all around it. (So, it’s like a skinny donut that is mostly hole, centered around the z-axis) (a) What is  in terms of M and R? What is the direction of the gravitational eld generated by this mass distribution at a point in space a distance z above the center of the donut, i.e. at (0; 0; z) Explain your reasoning for the direction carefully, try not to simply \wave your hands.” (The answer is extremely intuitive, but can you justify that it is correct?) (b) Compute the gravitational eld, ~g, at the point (0; 0; z) by directly integrating Newton’s law of gravity, summing over all in nitesimal \chunks” of mass along the loop. (c) Compute the gravitational potential at the point (0; 0; z) by directly integrating ?Gdm=r, sum- ming over all in nitesimal \chunks” dm along the loop. Then, take the z-component of the gradient of this potential to check that you agree with your result from the previous part. (d) In the two separate limits z << R and z >> R, Taylor expand your g- eld (in the z-direction)out only to the rst non-zero term, and convince us that both limits make good physical sense. (e) Can you use Gauss’ law to gure out the gravitational potential at the point (0; 0; z)? (If so, do it and check your previous answers. If not, why not?) Extra credit: If you place a small mass a small distance z away from the center, use your Taylor limit for z << R above to write a simple ODE for the equation of motion. Solve it, and discuss the motion Back to Problem List 4 5 { GRAVITATION Last Updated: July 16, 2012 5.4 Gravitational force near a metal-cored planet surrounded by a gaseous cloud Given: Pollock { Spring 2011 Jupiter is composed of a dense spherical core (of liquid metallic hydrogen!) of radius Rc. It is sur- rounded by a spherical cloud of gaseous hydrogen of radius Rg, where Rg > Rc. Let’s assume that the core is of uniform density c and the gaseous cloud is also of uniform density g. What is the gravitational force on an object of mass m that is located at a radius r from the center of Jupiter? Note that you must consider the cases where the object is inside the core, within the gas layer, and outside of the planet. Back to Problem List 5 5 { GRAVITATION Last Updated: July 16, 2012 5.5 Sphere with linearly increasing mass density Given: Pollock { Spring 2011 A planet of mass M and radius R has a nonuniform density that varies with r, the distance from the center according to  = Ar for 0  r  R. (a) What is the constant A in terms of M and R? Does this density pro le strike you as physically plausible, or is just designed as a mathematical exercise? (Brie y, explain) (b) Determine the gravitational force on a satellite of mass m orbiting this planet. In words, please outline the method you plan to use for your solution. (Use the easiest method you can come up with!) In your calculation, you will need to argue that the magnitude of ~g(r; ; ) depends only on r. Be very explicit about this – how do you know that it doesn’t, in fact, depend on  or ? (c) Determine the gravitational force felt by a rock of mass m inside the planet, located at radius r < R. (If the method you use is di erent than in part b, explain why you switched. If not, just proceed!) Explicitly check your result for this part by considering the limits r ! 0 and r ! R. Back to Problem List 6 5 { GRAVITATION Last Updated: July 16, 2012 5.6 Jumping o Vesta Given: Pollock { Spring 2011 You are stranded on the surface of the asteroid Vesta. If the mass of the asteroid is M and its radius is R, how fast would you have to jump o its surface to be able to escape from its gravitational eld? (Your estimate should be based on parameters that characterize the asteroid, not parameters that describe your jumping ability.) Given your formula, look up the approximate mass and radius of the asteroid Vesta 3 and determine a numerical value of the escape velocity. Could you escape in this way? (Brie y, explain) If so, roughly how big in radius is the maximum the asteroid could be, for you to still escape this way? If not, estimate how much smaller an asteroid you would need, to escape from it in this way? Figure 1: Back to Problem List 7 5 { GRAVITATION Last Updated: July 16, 2012 5.7 Gravitational force between two massive rods Given: Pollock { Spring 2011 Consider two identical uniform rods of length L and mass m lying along the same line and having their closest points separated by a distance d as shown in the gure (a) Calculate the mutual force between these rods, both its direction and magnitude. (b) Now do several checks. First, make sure the units worked out (!) The, nd the magnitude of the force in the limit L ! 0. What do you expect? Brie y, discuss. Lastly, nd the magnitude of the force in the limit d ! 1 ? Again, is it what you expect? Brie y, discuss. Figure 2: Given: Pollock { Spring 2012 Determining the gravitational force between two rods: (a) Consider a thin, uniform rod of mass m and length L (and negligible other dimensions) lying on the x axis (from x=-L to 0), as shown in g 1a. Derive a formula for the gravitational eld \g" at any arbitrary point x to the right of the origin (but still on the x-axis!) due to this rod. (b) Now suppose a second rod of length L and mass m sits on the x axis as shown in g 1b, with the left edge a distance \d" away. Calculate the mutual gravitational force between these rods. (c) Let's do some checks! Show that the units work out in parts a and b. Find the magnitude of the force in part a, in the limit x >> L: What do you expect? Brie y, discuss! Finally, verify that your answer to part b gives what you expect in the limit d >> L. ( Hint: This is a bit harder! You need to consistently expand everything to second order, not just rst, because of some interesting cancellations) Fig 1a Fig 1b L m +x x=0 L x=0 x=d m Fig 1a Fig 1b L m +x x=0 L +x x=0 x=d L m m Back to Problem List 8 5 { GRAVITATION Last Updated: July 16, 2012 5.8 Potential energy { Check your answer! Given: Pollock { Spring 2011 On the last exam, we had a problem with a at ring, uniform mass per unit area of , inner radius of R, outer radius of 2R. A satellite (mass m) sat a distance z above the center of the ring. We asked for the gravitational potential energy, and the answer was U(z) = ?2Gm( p 4R2 + z2 ? p R2 + z2) (1) (a) If you are far from the disk (on the z axis), what do you expect for the formula for U(z)? (Don’t say \0″ – as usual, we want the functional form of U(z) as you move far away. Also, explicitly state what we mean by \far away”. (Please don’t compare something with units to something without units!) (b) Show explicitly that the formula above does indeed give precisely the functional dependence you expect. Back to Problem List 9 5 { GRAVITATION Last Updated: July 16, 2012 5.9 Ways of solving gravitational problems Given: Pollock { Spring 2011, Spring 2012 Infinite cylinder ρ=cr x z (a) Half-infinite line mass, uniform linear mass density, λ x (b) R z  P Figure 3: (a) An in nite cylinder of radius R centered on the z-axis, with non-uniform volume mass density  = cr, where r is the radius in cylindrical coordinates. (b) A half-in nite line of mass on the x-axis extending from x = 0 to x = +1, with uniform linear mass density . There are two general methods we use to solve gravitational problems (i.e. nd ~g given some distribution of mass). (a) Describe these two methods. We claim one of these methods is easiest to solve for ~g of mass distribution (a) above, and the other method is easiest to solve for ~g of the mass distribution (b) above. Which method goes with which mass distribution? Please justify your answer. (b) Find ~g of the mass distribution (a) above for any arbitrary point outside the cylinder. (c) Find the x component of the gravitational acceleration, gx, generated by the mass distribution labeled (b) above, at a point P a given distance z up the positive z-axis (as shown). Back to Problem List 10 5 { GRAVITATION Last Updated: July 16, 2012 5.10 Rod with linearly increasing mass density Given: Pollock { Spring 2012 Consider a very (in nitesimally!) thin but massive rod, length L (total mass M), centered around the origin, sitting along the x-axis. (So the left end is at (-L/2, 0,0) and the right end is at (+L/2,0,0) Assume the mass density  (which has units of kg/m)is not uniform, but instead varies linearly with distance from the origin, (x) = cjxj. (a) What is that constant \c” in terms of M and L? What is the direction of the gravitational eld generated by this mass distribution at a point in space a distance z above the center of the rod, i.e. at (0; 0; z) Explain your reasoning for the direction carefully, try not to simply \wave your hands.” (The answer is extremely intuitive, but can you justify that it is correct?) (b) Compute the gravitational eld, ~g, at the point (0; 0; z) by directly integrating Newton’s law of gravity, summing over all in nitesimal \chunks” of mass along the rod. (c) Compute the gravitational potential at the point (0; 0; z) by directly integrating ?Gdm=r, sum- ming over all in nitesimal \chunks” dm along the rod. Then, take the z-component of the gradient of this potential to check that you agree with your result from the previous part. (d) In the limit of large z what do you expect for the functional form for gravitational potential? (Hint: Don’t just say it goes to zero! It’s a rod of mass M, when you’re far away what does it look like? How does it go to zero?) What does \large z” mean here? Use the binomial (or Taylor) expansion to verify that your formula does indeed give exactly what you expect. (Hint: you cannot Taylor expand in something BIG, you have to Taylor expand in something small.) (e) Can you use Gauss’ law to gure out the gravitational potential at the point (0; 0; z)? (If so, do it and check your previous answers. If not, why not?) Back to Problem List 11 5 { GRAVITATION Last Updated: July 16, 2012 5.11 Sphere with constant internal gravitational eld Given: Pollock { Spring 2012 (a) Imagine a planet of total mass M and radius R which has a nonuniform mass density that varies just with r, the distance from the center. For this (admittedly very unusual!) planet, suppose the gravitational eld strength inside the planet turns out to be independent of the radial distance within the sphere. Find the function describing the mass density  = (r) of this planet. (Your nal answer should be written in terms of the given constants.) (b) Now, determine the gravitational force on a satellite of mass m orbiting this planet at distance r > R. (Use the easiest method you can come up with!) Explain your work in words as well as formulas. For instance, in your calculation, you will need to argue that the magnitude of ~g(r; ; ) depends only on r. Be explicit about this – how do you know that it doesn’t, in fact, depend on  or ? (c) As a nal check, explicitly show that your solutions inside and outside the planet (parts a and b) are consistent when r = R. Please also comment on whether this density pro le strikes you as physically plausible, or is it just designed as a mathematical exercise? Defend your reasoning. Back to Problem List 12 5 { GRAVITATION Last Updated: July 16, 2012 5.12 Throwing a rock o the moon Given: Pollock { Spring 2012 Assuming that asteroids have roughly the same mass density as the moon, make an estimate of the largest asteroid that an astronaut could be standing on, and still have a chance of throwing a small object (with their arms, no machinery!) so that it completely escapes the asteroid’s gravitational eld. (This minimum speed is called \escape velocity”) Is the size you computed typical for asteroids in our solar system? Back to Problem List 13

Homework #8  Consider the veracity or falsehood of each of the following statements. For bonus, argue for those that you believe are true while providing a counterexample for those that you believe are false.  If the first and third rows of A are equal, then det A 0.  If P is a projection, then uCP if and only if Pu  u.  If P is a projection, and detP  0, then P  I .  If A has determinant 10, then 1 A has determinant 1 10 .  If B is invertible, 1 1 det(A B ) det A (detB) .  If P is a projection, and R  2P I , then 2 R  I .  If P is a projection, and P  I , then detP  0 .  Short Computations. All of the following do not involve long computations:  Suppose 1 2 1 5 1 8 A                  and 1 9 2 4 3 1 A                   . Compute 7 13 19 A         .  Compute               0 8 7 1 0 2 3 4 5 3 0 9 2 0 0 0 3 0 0 0 1 9 3 2 0 det .  Use Cramer’s Rule to find 5 x (hint: you do not need your calculator). 1 2 3 4 5 5x 2x 8x x 3x 13 1 3 3x 5x 0 1 3 5 3x 3x 3x 9 1 2 3 5 3x 2x x 2x 7 1 3 x 4x 0 Let A 1 2 3 4 1 3 4 6 2 5 13 15 4 10 15 31 . Given is that det A  61. Do the following:  1 1 2 4 2 3 5 10 3 4 13 15 4 6 15 31 det  det2A  1 3 4 6 2 4 6 8 2 5 13 15 4 10 15 31 det  1 3 4 6 2 5 13 15 4 10 15 31 1 2 3 4 det  Consider the matrix A  0 1 0 0 0 0 1 0 0 0 0 1 1 2 2 1           . Use row (or column) expansion to compute det(xI A) .  The matrix 4 1 1 2 1 1 1 4 1 1 2 1 1 1 4 1 1 2 2 1 1 4 1 1 1 2 1 1 4 1 1 1 2 1 1 4 1 6 P is the projection matrix for the column space of matrix A. This matrix A is also known to be of full rank. Answer the following, giving reasons for your answers.  Find a transparent basis and the dimension for the column space of P.  Find a basis and the dimension for the column space of A .  What size is the matrix A ?  Find a transparent basis and the dimension for the null space of P.  Find a transparent basis and the dimension for the row space of P.  Find a basis and the dimension for the null space of A.  For which of the following b can you find a solution to the system Ax b ? This does not mean you should find a solution, only whether one could or not. 10 17 19 14 10 17 19 14 13 10 17 19 14 13 23 1 1 1 1 1 1 .  It is known that certain vector u is a solution to the system Ax c . Give all solutions to Ax c .  It is also known that 1 2 3 4 5 6 Ax does not have a solution. How would you change the constant vector so that there would be a solution? Extra Problems.  Fill in the blank with the best possible expression to complete the sentence truthfully. Only that one will be counted correct. 1. matrix with two equal columns will have zero determinant. 1 2 3 Some Every No 2. If A is invertible, then A commute with its inverse. 1 2 3 must always can will not 3. If A is 6  9 , then the columns of A be linearly independent. While in AT , the columns be linearly independent. 1 2 3 can have to cannot 4. Let A be square, and suppose Ax  0 has a nontrivial solution. Then detA equal 0. 1 2 3 may cannot must 5. Let A and B be 3 3. Then det (AB) equal det(A)det(B) . 1 2 3 could must couldn’t 6. Let A be square and suppose detA  0. Then have an inverse 1 2 3 will not may must always 7. Let A and B be 2  2 . Then det (A B) equal det(A)  det(B) . 1 2 3 could must could not 8. exist a 6  6 matrix all of whose entries are whole numbers and its determinant is 2 5 . 1 2 3 There does There does not There might Bonus: Consider the matrix 0 0 1 0 2 0 n 0 . Give its determinant as a function of n.

Homework #8  Consider the veracity or falsehood of each of the following statements. For bonus, argue for those that you believe are true while providing a counterexample for those that you believe are false.  If the first and third rows of A are equal, then det A 0.  If P is a projection, then uCP if and only if Pu  u.  If P is a projection, and detP  0, then P  I .  If A has determinant 10, then 1 A has determinant 1 10 .  If B is invertible, 1 1 det(A B ) det A (detB) .  If P is a projection, and R  2P I , then 2 R  I .  If P is a projection, and P  I , then detP  0 .  Short Computations. All of the following do not involve long computations:  Suppose 1 2 1 5 1 8 A                  and 1 9 2 4 3 1 A                   . Compute 7 13 19 A         .  Compute               0 8 7 1 0 2 3 4 5 3 0 9 2 0 0 0 3 0 0 0 1 9 3 2 0 det .  Use Cramer’s Rule to find 5 x (hint: you do not need your calculator). 1 2 3 4 5 5x 2x 8x x 3x 13 1 3 3x 5x 0 1 3 5 3x 3x 3x 9 1 2 3 5 3x 2x x 2x 7 1 3 x 4x 0 Let A 1 2 3 4 1 3 4 6 2 5 13 15 4 10 15 31 . Given is that det A  61. Do the following:  1 1 2 4 2 3 5 10 3 4 13 15 4 6 15 31 det  det2A  1 3 4 6 2 4 6 8 2 5 13 15 4 10 15 31 det  1 3 4 6 2 5 13 15 4 10 15 31 1 2 3 4 det  Consider the matrix A  0 1 0 0 0 0 1 0 0 0 0 1 1 2 2 1           . Use row (or column) expansion to compute det(xI A) .  The matrix 4 1 1 2 1 1 1 4 1 1 2 1 1 1 4 1 1 2 2 1 1 4 1 1 1 2 1 1 4 1 1 1 2 1 1 4 1 6 P is the projection matrix for the column space of matrix A. This matrix A is also known to be of full rank. Answer the following, giving reasons for your answers.  Find a transparent basis and the dimension for the column space of P.  Find a basis and the dimension for the column space of A .  What size is the matrix A ?  Find a transparent basis and the dimension for the null space of P.  Find a transparent basis and the dimension for the row space of P.  Find a basis and the dimension for the null space of A.  For which of the following b can you find a solution to the system Ax b ? This does not mean you should find a solution, only whether one could or not. 10 17 19 14 10 17 19 14 13 10 17 19 14 13 23 1 1 1 1 1 1 .  It is known that certain vector u is a solution to the system Ax c . Give all solutions to Ax c .  It is also known that 1 2 3 4 5 6 Ax does not have a solution. How would you change the constant vector so that there would be a solution? Extra Problems.  Fill in the blank with the best possible expression to complete the sentence truthfully. Only that one will be counted correct. 1. matrix with two equal columns will have zero determinant. 1 2 3 Some Every No 2. If A is invertible, then A commute with its inverse. 1 2 3 must always can will not 3. If A is 6  9 , then the columns of A be linearly independent. While in AT , the columns be linearly independent. 1 2 3 can have to cannot 4. Let A be square, and suppose Ax  0 has a nontrivial solution. Then detA equal 0. 1 2 3 may cannot must 5. Let A and B be 3 3. Then det (AB) equal det(A)det(B) . 1 2 3 could must couldn’t 6. Let A be square and suppose detA  0. Then have an inverse 1 2 3 will not may must always 7. Let A and B be 2  2 . Then det (A B) equal det(A)  det(B) . 1 2 3 could must could not 8. exist a 6  6 matrix all of whose entries are whole numbers and its determinant is 2 5 . 1 2 3 There does There does not There might Bonus: Consider the matrix 0 0 1 0 2 0 n 0 . Give its determinant as a function of n.

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Chapter 03 Homework Due: 11:59pm on Friday, May 23, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Components and Structure of the Atom Learning Goal: To specify the basic components of the atom and describe our modern conception of its structure. Part A The atom consists of three types of subatomic particles: protons, neutrons, and electrons. The electron is by far the lightest of the three, while the much heavier proton and neutron have masses very similar to each other. Two of the types of particles carry an electrical charge, while the third is neutral. Label the subatomic particles and appropriate charges by their relative locations. Identify the subatomic particles by dragging the appropriate labels to their respective targets. Hint 1. Which subatomic particles carry electric charge? Of the three subatomic particles, two carry equal but opposite charges. Select the two correct statements that match the subatomic particle with the appropriate charge. Check all that apply. ANSWER: Hint 2. Which subatomic particles are not found in the nucleus? Protons and electrons carry equal but opposite charges. Atomic nuclei are positively charged and are not composed of negatively charged particles. Which types of subatomic particles cannot be located within the nucleus? Select any that apply. ANSWER: ANSWER: The electron carries a positive charge. The proton carries a positive charge. The neutron carries a positive charge. The proton carries a negative charge. The electron carries a negative charge. The neutron carries a negative charge. neutrons electrons protons Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 1 of 14 5/21/2014 8:02 PM Correct This image represents the classical model of the atom proposed by Niels Bohr. Although this model has changed slightly as the result of modern scientific discoveries, it does help in understanding the relative locations of the subatomic particles in the atom. Notice that the protons and neutrons are bound in the nucleus, while the electrons are located in the space surrounding the nucleus. Part B Of the three types of subatomic particles, only neutrons do not carry charge. Protons carry a positive charge, and electrons carry a negative charge. Protons and neutrons are bound in the nucleus, while electrons orbit the nucleus. When the number of each type of subatomic particle in an atom changes, the characteristics defining the atom also change. Match the appropriate phrases with the type of subatomic particle that completes the defining characteristic. Match the words in the left column to the appropriate blanks in the sentences on the right. Make certain each sentence is complete before submitting your answer. Hint 1. What type of subatomic particle is lost or gained when an ion forms? For any atom of a given element to go from being neutral ( ) to being ionized ( ), what type of subatomic particle must be lost or gained? Select all that apply. ANSWER: Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 2 of 14 5/21/2014 8:02 PM Hint 2. What type of subatomic particle identifies an element? When identifying the element classification of a particular atom, which type of subatomic particle is used? ANSWER: ANSWER: Correct The number of each type of subatomic particle plays an important role in the characteristics of the atom. The general element classification (hydrogen, carbon, oxygen, etc.) is governed by the number of protons in the nucleus. If the number of protons changes in an atom, so does the type of element. The electrons are the only type of subatomic particle not in the nucleus. They orbit around the nucleus, bound by the electromagnetic force. When electrons are lost or gained by a neutral atom, the charge balance shifts, resulting in the atom becoming an ion. Ions can be either positive when electrons are lost or negative when electrons are gained. Part C In the classical view of the atom, Bohr pictured electrons orbiting the positively charged nucleus similar to how the planets orbit the Sun. While this picture was not entirely correct, it provides a good framework in which to make calculations about the energies of electrons. Different from the predictions of Newtonian mechanics, which allows any energy to be possible, Bohr described the electron orbits (now called orbitals) as having specific energies. Rank the following electron energy states according to their electron energies. Rank from highest to lowest energies. Hint 1. What are the definitions of orbital, ground state, and excited state? Define orbital, ground state, and excited state. loss of an electron loss of a proton loss of a neutron gain of an electron gain of a proton gain of a neutron electron proton neutron Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 3 of 14 5/21/2014 8:02 PM Match the words in the left column to the appropriate blanks in the sentences on the right. Make certain each sentence is complete before submitting your answer. ANSWER: Hint 2. How does the state change when an electron absorbs energy? Electrons can absorb energy either from light radiation or from collisions with other atoms. If an electron is in the first excited energy state and absorbs enough energy to go to the next higher energy state, into what state will the electron transition? ANSWER: ANSWER: the ground state the second excited state the third excited state Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 4 of 14 5/21/2014 8:02 PM Correct Excited states refer to the energy of an electron. The higher the state, the higher the energy of the electron. The electron energies of each orbital are fixed. The energy required for an electron to transition between each orbital is an exact value, corresponding to the difference between the orbital energies. Any energy more or less than these precise differences cannot be used by the electron to make a transition; only the energies equal to the full values can induce a transition. Part D The Bohr model accounted for most of the general characteristics of the atom. However, the modern model based on quantum mechanics explains that, although the energy of each orbital is fixed, the orbital radius is actually an average distance. The result is a “cloud” where the electron is most likely to be located. The following is an image of an atom of hydrogen, consisting of one proton, zero neutrons, and one electron. When an electron is excited to different energy levels, the radius from the nucleus also changes. Rank the following electron energy states according to the average distance of the electron from the nucleus. Rank from largest to smallest distances. Hint 1. What is the relationship between electron orbital distance and electron energy? Rank the following general electron energies from largest to smallest electron orbital distances. Rank from largest to smallest orbital distances. ANSWER: ANSWER: Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 5 of 14 5/21/2014 8:02 PM Correct Excited states refer to the energy state of an electron. The higher the state, the higher the energy and the greater the distance of the electron from the nucleus. Due to the attractive force between the negatively charged electron and the positively charged nucleus, the electron requires greater energies to overcome this attraction and achieve orbits at greater distances. Concept Review: The pH Scale Can you classify solutions as acidic, neutral, or basic? Part A Decide whether each label describes a solution that is acidic, neutral, or basic, and then drag it into the appropriate bin. ANSWER: Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 6 of 14 5/21/2014 8:02 PM Correct Activity: Carbohydrates Click here to complete this activity. Then answer the questions. Part A Glycogen is _____. ANSWER: Correct Animals store energy in the form of glycogen. a polysaccharide found in animals a source of saturated fat a polysaccharide found in plant cell walls the form in which plants store sugars a transport protein that carries oxygen Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 7 of 14 5/21/2014 8:02 PM Part B glucose + glucose —> _____ by _____. ANSWER: Correct Maltose is the disaccharide formed when two glucose molecules are linked by dehydration synthesis. Part C Which of these is a source of lactose? ANSWER: Correct Lactose is the sugar found in milk. Part D Which of these is a polysaccharide? ANSWER: Correct Cellulose is a carbohydrate composed of many monomers. Part E _____ is the most abundant organic compound on Earth. ANSWER: maltose + water … dehydration synthesis lactose + water … hydrolysis starch + water … dehydration synthesis sucrose + water … dehydration synthesis cellulose + water … hydrolysis potatoes sugar beets sugar cane starch milk sucrose lactose glucose galactose cellulose Cellulose Lactose Starch Glucose Glycogen Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 8 of 14 5/21/2014 8:02 PM Correct Cellulose, a component of plant cell walls, is the most abundant organic compound found on earth. Activity: Protein Structure Click here to complete this activity. Then answer the questions. Part A Proteins are polymers of _____. ANSWER: Correct Proteins are polymers of amino acids. Part B What type of bond joins the monomers in a protein’s primary structure? ANSWER: Correct The amino acids of a protein are linked by peptide bonds. Part C Which of these illustrates the secondary structure of a protein? ANSWER: nucleotides CH2O units glycerol hydrocarbons amino acids ionic hydrogen hydrophobic S—S peptide Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 9 of 14 5/21/2014 8:02 PM Correct Alpha helices and beta pleated sheets are characteristic of a protein’s secondary structure. Part D The secondary structure of a protein results from _____. ANSWER: Correct Electronegative oxygen and nitrogen atoms leave hydrogen atoms with partial positive charges. Part E Tertiary structure is NOT directly dependent on _____. ANSWER: bonds between sulfur atoms peptide bonds hydrogen bonds hydrophobic interactions ionic bonds hydrophobic interactions ionic bonds hydrogen bonds peptide bonds bonds between sulfur atoms Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 10 of 14 5/21/2014 8:02 PM Correct Peptide bonds link together the amino acids of a protein’s primary structure. Activity: Lipids Click here to complete this activity. Then answer the questions. Part A Which of these is NOT a lipid? ANSWER: Correct RNA is a nucleic acid Part B This figure is an example of a(n) _____. ANSWER: Correct The fatty acid tails lack double bonds. steroids phospholipid RNA cholesterol wax steroid unsaturated fat nucleic acid protein saturated fat Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 11 of 14 5/21/2014 8:02 PM Part C Which of these is a phospholipid? ANSWER: Correct Phospholipids are composed of a phosphate group, a glycerol, and fatty acids. Part D Which of these is rich in unsaturated fats? ANSWER: Correct Olive oil is a plant oil, and most plant oils are rich in unsaturated fats. Part E beef fat lard butter olive oil a fat that is solid at room temperature Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 12 of 14 5/21/2014 8:02 PM A function of cholesterol that does not harm health is its role _____. ANSWER: Correct Cholesterol is an important component of animal cell membranes. Concept Review: Types of Macromolecules Can you identify characteristics of proteins, nucleic acids, and carbohydrates? Part A Decide whether each label describes proteins, nucleic acids, or carbohydrates, and then drag it into the appropriate bin. ANSWER: Correct Concept Review: Earth’s Interior Layers Can you identify characteristics of Earth’s interior layers? Part A Drag the labels to the appropriate targets. ANSWER: as a component of animal cell membranes in calcium and phosphate metabolism All of cholesterol’s effects cause the body harm. as the most abundant male sex hormone as the primary female sex hormone Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 13 of 14 5/21/2014 8:02 PM Correct Score Summary: Your score on this assignment is 99.6%. You received 31.87 out of a possible total of 32 points. Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 14 of 14 5/21/2014 8:02 PM

Chapter 03 Homework Due: 11:59pm on Friday, May 23, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Components and Structure of the Atom Learning Goal: To specify the basic components of the atom and describe our modern conception of its structure. Part A The atom consists of three types of subatomic particles: protons, neutrons, and electrons. The electron is by far the lightest of the three, while the much heavier proton and neutron have masses very similar to each other. Two of the types of particles carry an electrical charge, while the third is neutral. Label the subatomic particles and appropriate charges by their relative locations. Identify the subatomic particles by dragging the appropriate labels to their respective targets. Hint 1. Which subatomic particles carry electric charge? Of the three subatomic particles, two carry equal but opposite charges. Select the two correct statements that match the subatomic particle with the appropriate charge. Check all that apply. ANSWER: Hint 2. Which subatomic particles are not found in the nucleus? Protons and electrons carry equal but opposite charges. Atomic nuclei are positively charged and are not composed of negatively charged particles. Which types of subatomic particles cannot be located within the nucleus? Select any that apply. ANSWER: ANSWER: The electron carries a positive charge. The proton carries a positive charge. The neutron carries a positive charge. The proton carries a negative charge. The electron carries a negative charge. The neutron carries a negative charge. neutrons electrons protons Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 1 of 14 5/21/2014 8:02 PM Correct This image represents the classical model of the atom proposed by Niels Bohr. Although this model has changed slightly as the result of modern scientific discoveries, it does help in understanding the relative locations of the subatomic particles in the atom. Notice that the protons and neutrons are bound in the nucleus, while the electrons are located in the space surrounding the nucleus. Part B Of the three types of subatomic particles, only neutrons do not carry charge. Protons carry a positive charge, and electrons carry a negative charge. Protons and neutrons are bound in the nucleus, while electrons orbit the nucleus. When the number of each type of subatomic particle in an atom changes, the characteristics defining the atom also change. Match the appropriate phrases with the type of subatomic particle that completes the defining characteristic. Match the words in the left column to the appropriate blanks in the sentences on the right. Make certain each sentence is complete before submitting your answer. Hint 1. What type of subatomic particle is lost or gained when an ion forms? For any atom of a given element to go from being neutral ( ) to being ionized ( ), what type of subatomic particle must be lost or gained? Select all that apply. ANSWER: Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 2 of 14 5/21/2014 8:02 PM Hint 2. What type of subatomic particle identifies an element? When identifying the element classification of a particular atom, which type of subatomic particle is used? ANSWER: ANSWER: Correct The number of each type of subatomic particle plays an important role in the characteristics of the atom. The general element classification (hydrogen, carbon, oxygen, etc.) is governed by the number of protons in the nucleus. If the number of protons changes in an atom, so does the type of element. The electrons are the only type of subatomic particle not in the nucleus. They orbit around the nucleus, bound by the electromagnetic force. When electrons are lost or gained by a neutral atom, the charge balance shifts, resulting in the atom becoming an ion. Ions can be either positive when electrons are lost or negative when electrons are gained. Part C In the classical view of the atom, Bohr pictured electrons orbiting the positively charged nucleus similar to how the planets orbit the Sun. While this picture was not entirely correct, it provides a good framework in which to make calculations about the energies of electrons. Different from the predictions of Newtonian mechanics, which allows any energy to be possible, Bohr described the electron orbits (now called orbitals) as having specific energies. Rank the following electron energy states according to their electron energies. Rank from highest to lowest energies. Hint 1. What are the definitions of orbital, ground state, and excited state? Define orbital, ground state, and excited state. loss of an electron loss of a proton loss of a neutron gain of an electron gain of a proton gain of a neutron electron proton neutron Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 3 of 14 5/21/2014 8:02 PM Match the words in the left column to the appropriate blanks in the sentences on the right. Make certain each sentence is complete before submitting your answer. ANSWER: Hint 2. How does the state change when an electron absorbs energy? Electrons can absorb energy either from light radiation or from collisions with other atoms. If an electron is in the first excited energy state and absorbs enough energy to go to the next higher energy state, into what state will the electron transition? ANSWER: ANSWER: the ground state the second excited state the third excited state Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 4 of 14 5/21/2014 8:02 PM Correct Excited states refer to the energy of an electron. The higher the state, the higher the energy of the electron. The electron energies of each orbital are fixed. The energy required for an electron to transition between each orbital is an exact value, corresponding to the difference between the orbital energies. Any energy more or less than these precise differences cannot be used by the electron to make a transition; only the energies equal to the full values can induce a transition. Part D The Bohr model accounted for most of the general characteristics of the atom. However, the modern model based on quantum mechanics explains that, although the energy of each orbital is fixed, the orbital radius is actually an average distance. The result is a “cloud” where the electron is most likely to be located. The following is an image of an atom of hydrogen, consisting of one proton, zero neutrons, and one electron. When an electron is excited to different energy levels, the radius from the nucleus also changes. Rank the following electron energy states according to the average distance of the electron from the nucleus. Rank from largest to smallest distances. Hint 1. What is the relationship between electron orbital distance and electron energy? Rank the following general electron energies from largest to smallest electron orbital distances. Rank from largest to smallest orbital distances. ANSWER: ANSWER: Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 5 of 14 5/21/2014 8:02 PM Correct Excited states refer to the energy state of an electron. The higher the state, the higher the energy and the greater the distance of the electron from the nucleus. Due to the attractive force between the negatively charged electron and the positively charged nucleus, the electron requires greater energies to overcome this attraction and achieve orbits at greater distances. Concept Review: The pH Scale Can you classify solutions as acidic, neutral, or basic? Part A Decide whether each label describes a solution that is acidic, neutral, or basic, and then drag it into the appropriate bin. ANSWER: Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 6 of 14 5/21/2014 8:02 PM Correct Activity: Carbohydrates Click here to complete this activity. Then answer the questions. Part A Glycogen is _____. ANSWER: Correct Animals store energy in the form of glycogen. a polysaccharide found in animals a source of saturated fat a polysaccharide found in plant cell walls the form in which plants store sugars a transport protein that carries oxygen Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 7 of 14 5/21/2014 8:02 PM Part B glucose + glucose —> _____ by _____. ANSWER: Correct Maltose is the disaccharide formed when two glucose molecules are linked by dehydration synthesis. Part C Which of these is a source of lactose? ANSWER: Correct Lactose is the sugar found in milk. Part D Which of these is a polysaccharide? ANSWER: Correct Cellulose is a carbohydrate composed of many monomers. Part E _____ is the most abundant organic compound on Earth. ANSWER: maltose + water … dehydration synthesis lactose + water … hydrolysis starch + water … dehydration synthesis sucrose + water … dehydration synthesis cellulose + water … hydrolysis potatoes sugar beets sugar cane starch milk sucrose lactose glucose galactose cellulose Cellulose Lactose Starch Glucose Glycogen Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 8 of 14 5/21/2014 8:02 PM Correct Cellulose, a component of plant cell walls, is the most abundant organic compound found on earth. Activity: Protein Structure Click here to complete this activity. Then answer the questions. Part A Proteins are polymers of _____. ANSWER: Correct Proteins are polymers of amino acids. Part B What type of bond joins the monomers in a protein’s primary structure? ANSWER: Correct The amino acids of a protein are linked by peptide bonds. Part C Which of these illustrates the secondary structure of a protein? ANSWER: nucleotides CH2O units glycerol hydrocarbons amino acids ionic hydrogen hydrophobic S—S peptide Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 9 of 14 5/21/2014 8:02 PM Correct Alpha helices and beta pleated sheets are characteristic of a protein’s secondary structure. Part D The secondary structure of a protein results from _____. ANSWER: Correct Electronegative oxygen and nitrogen atoms leave hydrogen atoms with partial positive charges. Part E Tertiary structure is NOT directly dependent on _____. ANSWER: bonds between sulfur atoms peptide bonds hydrogen bonds hydrophobic interactions ionic bonds hydrophobic interactions ionic bonds hydrogen bonds peptide bonds bonds between sulfur atoms Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 10 of 14 5/21/2014 8:02 PM Correct Peptide bonds link together the amino acids of a protein’s primary structure. Activity: Lipids Click here to complete this activity. Then answer the questions. Part A Which of these is NOT a lipid? ANSWER: Correct RNA is a nucleic acid Part B This figure is an example of a(n) _____. ANSWER: Correct The fatty acid tails lack double bonds. steroids phospholipid RNA cholesterol wax steroid unsaturated fat nucleic acid protein saturated fat Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 11 of 14 5/21/2014 8:02 PM Part C Which of these is a phospholipid? ANSWER: Correct Phospholipids are composed of a phosphate group, a glycerol, and fatty acids. Part D Which of these is rich in unsaturated fats? ANSWER: Correct Olive oil is a plant oil, and most plant oils are rich in unsaturated fats. Part E beef fat lard butter olive oil a fat that is solid at room temperature Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 12 of 14 5/21/2014 8:02 PM A function of cholesterol that does not harm health is its role _____. ANSWER: Correct Cholesterol is an important component of animal cell membranes. Concept Review: Types of Macromolecules Can you identify characteristics of proteins, nucleic acids, and carbohydrates? Part A Decide whether each label describes proteins, nucleic acids, or carbohydrates, and then drag it into the appropriate bin. ANSWER: Correct Concept Review: Earth’s Interior Layers Can you identify characteristics of Earth’s interior layers? Part A Drag the labels to the appropriate targets. ANSWER: as a component of animal cell membranes in calcium and phosphate metabolism All of cholesterol’s effects cause the body harm. as the most abundant male sex hormone as the primary female sex hormone Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 13 of 14 5/21/2014 8:02 PM Correct Score Summary: Your score on this assignment is 99.6%. You received 31.87 out of a possible total of 32 points. Chapter 03 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 14 of 14 5/21/2014 8:02 PM

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

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

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1. Isolate GT1 in the equation ΔGT2 T2 − ΔGT1 T1 = ΔH ( 1 T2 − 1 T1 ). 2. True or False? ln (a + b) = ln a + ln b … prove your answer numerically. 3. What is the base-e logarithm of 250, ln 250? Prove that your result works numerically. 4. Solve for x in the following equation: e–ax = 1/T. 5. Simplify the right side of the function y = eAeC/eT and then use the ln to solve for T. 6. At what values of ϕ are the functions sin(ϕ) or cos(ϕ) equal to 0? At what values are they each equal to 1? 7. Linearize the following equation to find m and K from the slope and intercept: v = m[X]/(K + [X]). 8. Find the 1st and 2nd derivatives of y(x) = 3×4 – 2×2 + 15. 9. Identify the locations of minima and maxima for the function given in the problem above. 10. Find the derivative of the function y(x) = 3 ln (2×2). Ψ(x), [Ψ(x)]2, Ψ’(x), and Ψ’’(x). 12. Integrate 3/x from to 1 to 3. 13. What is the integral of 3×2 – 2x + 4 between -1 and 1? 14. Integrate the following function from 2 to 4: y = 3 cos x + x/2. (Don’t forget to check your calculator settings! Is it set for degrees or radians?) 15. Isolate like terms and integrate both sides of your resulting differential equation: dy/dx = – 4 x-1 y-3.

1. Isolate GT1 in the equation ΔGT2 T2 − ΔGT1 T1 = ΔH ( 1 T2 − 1 T1 ). 2. True or False? ln (a + b) = ln a + ln b … prove your answer numerically. 3. What is the base-e logarithm of 250, ln 250? Prove that your result works numerically. 4. Solve for x in the following equation: e–ax = 1/T. 5. Simplify the right side of the function y = eAeC/eT and then use the ln to solve for T. 6. At what values of ϕ are the functions sin(ϕ) or cos(ϕ) equal to 0? At what values are they each equal to 1? 7. Linearize the following equation to find m and K from the slope and intercept: v = m[X]/(K + [X]). 8. Find the 1st and 2nd derivatives of y(x) = 3×4 – 2×2 + 15. 9. Identify the locations of minima and maxima for the function given in the problem above. 10. Find the derivative of the function y(x) = 3 ln (2×2). Ψ(x), [Ψ(x)]2, Ψ’(x), and Ψ’’(x). 12. Integrate 3/x from to 1 to 3. 13. What is the integral of 3×2 – 2x + 4 between -1 and 1? 14. Integrate the following function from 2 to 4: y = 3 cos x + x/2. (Don’t forget to check your calculator settings! Is it set for degrees or radians?) 15. Isolate like terms and integrate both sides of your resulting differential equation: dy/dx = – 4 x-1 y-3.

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1. The reaction time of a driver to visual stimulus is normally distributed with a mean of 0.2 seconds and a standard deviation of 0.1 seconds. 1‐1. (2 points) What is the probability that a reaction requires more than 0.5 seconds? 1‐2. (2 points) What is the probability that a reaction requires between 0.4 and 0.5 seconds? 1‐3. (2 points) What is the reaction time that is exceeded 95% of the time? 2. Spherical Uniform Distribution (Google! You do not have to explain why): 2‐1. (2 points) How can we pick a set of random points uniformly distributed on the unit circle x12 + x 2=1? 2‐2. (2 points) How can we pick a set of random points uniformly distributed on the 4‐dimensional unit 2 2 2 2 2 sphere x1 + x2 + x3 + x4 + x5 =1? 3. The random variable X has a binomial distribution with n = 19 and p = 0.4. Determine the following probabilities. (You may use computer. But, you have to show the formula.) 3‐1. (2 points) P(X ≤ 12) 3‐2. (2 points) P(X ≥ 18) 3‐3. (2 points) P(13 ≤ X < 15) 4. (2 points) Show the mean and the variance of the triangular distribution with lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b. (You must show why.) 5. (2 points) An electronic office product contains 5000 electronic components. Assume that the probability that each component operates without failure during the useful life of the product is 0.999, and assume that the components fail independently. Approximate the probability that 10 or more of the original 5000 components fail during the useful life of the product. 6. Consider the following system made up of functional components in parallel and series. C2 0.80 C1 0.90 C4 0.95 C3 0.85 6‐1. (2 points) What is the probability that the system operates? 6‐2. (2 points) What is the probability that the system fails due to the components in series? Assume parallel components do not fail. 6‐3. (2 points) What is the probability that the system fails due to the components in parallel? Assume series components do not fail. 6‐4. (2 points) Compute and compare the probabilities that the system fails when the probability that component C1 functions is improved to a value of 0.95 and when the probability that component C2 functions is improved to a value of 0.85. Which improvement increases the system reliability more? 7. (2 points) Suppose that the joint distribution of X and Y has probability density function f(x, y) = 0.25xy for 0 < x < 2 and 0 < y < 2. Compute V(2X + 3Y). (Show all your work.)

1. The reaction time of a driver to visual stimulus is normally distributed with a mean of 0.2 seconds and a standard deviation of 0.1 seconds. 1‐1. (2 points) What is the probability that a reaction requires more than 0.5 seconds? 1‐2. (2 points) What is the probability that a reaction requires between 0.4 and 0.5 seconds? 1‐3. (2 points) What is the reaction time that is exceeded 95% of the time? 2. Spherical Uniform Distribution (Google! You do not have to explain why): 2‐1. (2 points) How can we pick a set of random points uniformly distributed on the unit circle x12 + x 2=1? 2‐2. (2 points) How can we pick a set of random points uniformly distributed on the 4‐dimensional unit 2 2 2 2 2 sphere x1 + x2 + x3 + x4 + x5 =1? 3. The random variable X has a binomial distribution with n = 19 and p = 0.4. Determine the following probabilities. (You may use computer. But, you have to show the formula.) 3‐1. (2 points) P(X ≤ 12) 3‐2. (2 points) P(X ≥ 18) 3‐3. (2 points) P(13 ≤ X < 15) 4. (2 points) Show the mean and the variance of the triangular distribution with lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b. (You must show why.) 5. (2 points) An electronic office product contains 5000 electronic components. Assume that the probability that each component operates without failure during the useful life of the product is 0.999, and assume that the components fail independently. Approximate the probability that 10 or more of the original 5000 components fail during the useful life of the product. 6. Consider the following system made up of functional components in parallel and series. C2 0.80 C1 0.90 C4 0.95 C3 0.85 6‐1. (2 points) What is the probability that the system operates? 6‐2. (2 points) What is the probability that the system fails due to the components in series? Assume parallel components do not fail. 6‐3. (2 points) What is the probability that the system fails due to the components in parallel? Assume series components do not fail. 6‐4. (2 points) Compute and compare the probabilities that the system fails when the probability that component C1 functions is improved to a value of 0.95 and when the probability that component C2 functions is improved to a value of 0.85. Which improvement increases the system reliability more? 7. (2 points) Suppose that the joint distribution of X and Y has probability density function f(x, y) = 0.25xy for 0 < x < 2 and 0 < y < 2. Compute V(2X + 3Y). (Show all your work.)

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A company produces bag, each of which could be sold for$150. It has a fixed monthly cost of $13,000 and a variable cost of $20 per bag produced. A. Determine the company monthly total cost function B. determine the company’s monthly revenue function C determine the company’s monthly total profit function D how many bag must the company produce to break even E how many bags must the company produce to generate $26,000 in profit? Compute A log 100 B log 1000 C log2 8 D log5. 25 E log 0.01 F log 1 G loga a

A company produces bag, each of which could be sold for$150. It has a fixed monthly cost of $13,000 and a variable cost of $20 per bag produced. A. Determine the company monthly total cost function B. determine the company’s monthly revenue function C determine the company’s monthly total profit function D how many bag must the company produce to break even E how many bags must the company produce to generate $26,000 in profit? Compute A log 100 B log 1000 C log2 8 D log5. 25 E log 0.01 F log 1 G loga a