Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

# Homework Problems

Derive a formula expressing the kinetic energy of an object in terms of its momentum and mass.
Two people in a rowboat wish to move around without causing the boat to move. What should be true about their total momentum? Explain.
A learjet traveling due east at 300 mi/hr collides with a jumbo jet which was heading southwest at 150 mi/hr. The jumbo jet's mass is 5.0 times greater than that of the learjet. When they collide, the learjet sticks into the fuselage of the jumbo jet, and they fall to earth together. Their engines stop functioning immediately after the collision. On a map, what will be the direction from the location of the collision to the place where the wreckage hits the ground? (Give an angle.)
A bullet leaves the barrel of a gun with a kinetic energy of 90 J. The gun barrel is 50 cm long. The gun has a mass of 4 kg, the bullet 10 g.

(a) Find the bullet's final velocity.

(b) Find the bullet's final momentum.

(c) Find the momentum of the recoiling gun.

(d) Find the kinetic energy of the recoiling gun, and explain why the recoiling gun does not kill the shooter.

The graph shows the force, in meganewtons, exerted by a rocket engine on the rocket as a function of time. If the rocket's mass is 4000 kg, at what speed is the rocket moving when the engine stops firing? Assume it goes straight up, and neglect the force of gravity, which is much less than a meganewton.

Cosmic rays are particles from outer space, mostly protons and atomic nuclei, that are continually bombarding the earth. Most of them, although they are moving extremely fast, have no discernible effect even if they hit your body, because their masses are so small. Their energies vary, however, and a very small minority of them have extremely large energies. In some cases the energy is as much as several Joules, which is comparable to the KE of a well thrown rock! If you are in a plane at a high altitude and are so incredibly unlucky as to be hit by one of these rare ultra-high-energy cosmic rays, what would you notice, the momentum imparted to your body, the energy dissipated in your body as heat, or both? Base your conclusions on numerical estimates, not just random speculation. (At these high speeds, one should really take into account the deviations from Newtonian physics described by Einstein's special theory of relativity. Don't worry about that, though.)
Show that for a body made up of many equal masses, the equation for the center of mass becomes a simple average of all the positions of the masses.

The figure shows a view from above of a collision about to happen between two air hockey pucks sliding without friction. They have the same speed, vi, before the collision, but the big puck is 2.3 times more massive than the small one. Their sides have sticky stuff on them, so when they collide, they will stick together. At what angle will they emerge from the collision? In addition to giving a numerical answer, please indicate by drawing on the figure how your angle is defined.

Solution, p. 160
A flexible rope of mass m and length L slides without friction over the edge of a table. Let x be the length of the rope that is hanging over the edge at a given moment in time.

(a) Show that x satisfies the equation of motion d2x/dt2 = gx/L. [Hint: Use F = dp/dt, which allows you to handle the two parts of the rope separately even though mass is moving out of one part and into the other.]

(b) Give a physical explanation for the fact that a larger value of x on the right-hand side of the equation leads to a greater value of the acceleration on the left side.

(c) When we take the second derivative of the function x(t) we are supposed to get essentially the same function back again, except for a constant out in front. The function ex has the property that it is unchanged by differentiation, so it is reasonable to look for solutions to this problem that are of the form x = bect, where b and c are constants. Show that this does indeed provide a solution for two specific values of c (and for any value of b).

(d) Show that the sum of any two solutions to the equation of motion is also a solution.

(e) Find the solution for the case where the rope starts at rest at t = 0 with some nonzero value of x. R ?

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10 A very massive object with velocity v collides head-on with an object at rest whose mass is very small. No kinetic energy is converted into other forms. Prove that the low-mass object recoils with velocity 2v. [Hint: Use the center-of-mass frame of reference.]
11 When the contents of a refrigerator cool down, the changed molecular speeds imply changes in both momentum and energy. Why, then, does a fridge transfer power through its radiator coils, but not force?
Solution, p. 160
12 A 10-kg bowling ball moving at 2.0 m/s hits a 1.0-kg bowling pin, which is initially at rest. The other pins are all gone already, and the collision is head-on, so that the motion is one-dimensional. Assume that negligible amounts of heat and sound are produced. Find the velocity of the pin immediately after the collision.
13 A rocket ejects exhaust with an exhaust velocity u. The rate at which the exhaust mass is used (mass per unit time) is b. We assume that the rocket accelerates in a straight line starting from rest, and that no external forces act on it. Let the rocket's initial mass (fuel plus the body and payload) be mi, and mf be its final mass, after all the fuel is used up. (a) Find the rocket's final velocity, v, in terms of u, mi, and mf . (b) A typical exhaust velocity for chemical rocket engines is 4000 m/s. Estimate the initial mass of a rocket that could accelerate a one-ton payload to 10% of the speed of light, and show that this design won't work. (For the sake of the estimate, ignore the mass of the fuel tanks.)
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14 A firework shoots up into the air, and just before it explodes it has a certain momentum and kinetic energy. What can you say about the momenta and kinetic energies of the pieces immediately after the explosion? [Based on a problem from PSSC Physics.]
Solution, p. 160
15 Suppose a system consisting of pointlike particles has a total kinetic energy Kcm measured in the center-of-mass frame of reference. Since they are pointlike, they cannot have any energy due to internal motion.

(a) Prove that in a different frame of reference, moving with velocity u relative to the center-of-mass frame, the total kinetic energy equals Kcm +M|u|2/2, where M is the total mass. [Hint: You can save yourself a lot of writing if you express the total kinetic energy using the dot product.]

(b) Use this to prove that if energy is conserved in one frame of reference, then it is conserved in every frame of reference. The total 100 Chapter 4 Conservation of Momentum energy equals the total kinetic energy plus the sum of the potential energies due to the particles' interactions with each other, which we assume depends only on the distance between particles. [For a simpler numerical example, see problem 13 in ch. 1.] ?

Solution, p. 161 *
16 The big difference between the equations for momentum and kinetic energy is that one is proportional to v and one to v2. Both, however, are proportional to m. Suppose someone tells you that there's a third quantity, funkosity, defined as f = m2v, and that funkosity is conserved. How do you know your leg is being pulled?
Solution, p. 161
17 A mass m moving at velocity v collides with a stationary target having the same mass m. Find the maximum amount of energy that can be released as heat and sound.

Last Update: 2009-06-21