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?
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
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 ?
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.]
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
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.
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.)
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
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
(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 *
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
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.