Two cars with different masses each have the same kinetic
(a) If both cars have the same brakes, capable of supplying
the same force, how will the stopping distances compare? Explain.
(b) Compare the times required for the cars to stop.
In each of the following situations, is the work being done
positive, negative, or zero?
(a) a bull paws the ground;
(b) a fishing
boat pulls a net through the water behind it;
(c) the water resists
the motion of the net through it;
(d) you stand behind a pickup
truck and lower a bale of hay from the truck's bed to the ground.
[Based on a problem by Serway and Faughn.]
In the earth's atmosphere, the molecules are constantly moving
around. Because temperature is a measure of kinetic energy per
molecule, the average kinetic energy of each type of molecule is the
same, e.g., the average KE of the O2 molecules is the same as the
average KE of the N2 molecules.
(a) If the mass of an O2 molecule
is eight times greater than that of a He atom, what is the ratio of
their average speeds? Which way is the ratio, i.e., which is typically
(b) Use your result from part a to explain why any
helium occurring naturally in the atmosphere has long since escaped
into outer space, never to return. (Helium is obtained commercially
by extracting it from rocks.) You may want to do problem 20 first,
Weiping lifts a rock with a weight of 1.0 N through a height of
1.0 m, and then lowers it back down to the starting point. Bubba
pushes a table 1.0 m across the floor at constant speed, requiring
a force of 1.0 N, and then pushes it back to where it started.
Compare the total work done by Weiping and Bubba.
that your answers to part a make sense, using the definition of work:
work is the transfer of energy. In your answer, you'll need to discuss
what specific type of energy is involved in each case.
In one of his more flamboyant moments, Galileo wrote Who
does not know that a horse falling from a height of three or four
cubits will break his bones, while a dog falling from the same height
or a cat from a height of eight or ten cubits will suffer no injury?
Equally harmless would be the fall of a grasshopper from a tower or
the fall of an ant from the distance of the moon. Find the speed
of an ant that falls to earth from the distance of the moon at the
moment when it is about to enter the atmosphere. Assume it is
released from a point that is not actually near the moon, so the
moon's gravity is negligible.
(a) The crew of an 18th century warship is raising the anchor.
The anchor has a mass of 5000 kg. The water is 30 m deep. The
chain to which the anchor is attached has a mass per unit length of
150 kg/m. Before they start raising the anchor, what is the total
weight of the anchor plus the portion of the chain hanging out of the
ship? (Assume that the buoyancy of the anchor and is negligible.)
(b) After they have raised the anchor by 1 m, what is the weight
they are raising?
(c) Define y = 0 when the anchor is resting on the bottom, and
y = +30 m when it has been raised up to the ship. Draw a graph
of the force the crew has to exert to raise the anchor and chain, as
a function of y. (Assume that they are raising it slowly, so water
resistance is negligible.) It will not be a constant! Now find the
area under the graph, and determine the work done by the crew in
raising the anchor, in joules.
(d) Convert your answer from (c) into units of kcal.
A cylinder from
the 1965 Rambler's engine. The
piston is shown in its pushed out
position. The two bulges at the
top are for the valves that let fresh
air-gas mixture in. Based on a
figure from Motor Service's Automotive
In the power stroke of a car's gasoline engine, the fuel-air mixture
is ignited by the spark plug, explodes, and pushes the piston
out. The exploding mixture's force on the piston head is greatest
at the beginning of the explosion, and decreases as the mixture expands.
It can be approximated by F = a/x, where x is the distance
from the cylinder to the piston head, and a is a constant with units
of N.m. (Actually a/x1.4 would be more accurate, but the problem
works out more nicely with a/x!) The piston begins its stroke at
x = x1, and ends at x = x2. The 1965 Rambler had six cylinders,
each with a = 220 N·m, x1 = 1.2 cm, and x2 = 10.2 cm.
(a) Draw a neat, accurate graph of F vs x, on graph paper.
(b) From the area under the curve, derive the amount of work done
in one stroke by one cylinder.
(c) Assume the engine is running at 4800 r.p.m., so that during
one minute, each of the six cylinders performs 2400 power strokes.
(Power strokes only happen every other revolution.) Find the engine's
power, in units of horsepower (1 hp=746 W).
(d) The compression ratio of an engine is defined as x2/x1. Explain
in words why the car's power would be exactly the same if x1 and
x2 were, say, halved or tripled, maintaining the same compression
ratio of 8.5. Explain why this would not quite be true with the more
realistic force equation F = a/x1.4.
The magnitude of the force between two magnets separated
by a distance r can be approximated as kr-3 for large values of r.
The constant k depends on the strengths of the magnets and the
relative orientations of their north and south poles. Two magnets
are released on a slippery surface at an initial distance ri, and begin
sliding towards each other. What will be the total kinetic energy
of the two magnets when they reach a final distance rf ? (Ignore
A car starts from rest at t = 0, and starts speeding up with
(a) Find the car's kinetic energy in terms of
its mass, m, acceleration, a, and the time, t.
(b) Your answer in
the previous part also equals the amount of work, W, done from
t = 0 until time t. Take the derivative of the previous expression
to find the power expended by the car at time t.
(c) Suppose two
cars with the same mass both start from rest at the same time, but
one has twice as much acceleration as the other. At any moment,
how many times more power is being dissipated by the more quickly
accelerating car? (The answer is not 2.)
A space probe of mass m is dropped into a previously unexplored
spherical cloud of gas and dust, and accelerates toward
the center of the cloud under the influence of the cloud's gravity.
Measurements of its velocity allow its potential energy, U, to be
determined as a function of the distance r from the cloud's center.
The mass in the cloud is distributed in a spherically symmetric way,
so its density, ρ(r), depends only on r and not on the angular coordinates.
Show that by finding U(r), one can infer ρ(r) as follows:
A rail gun is a device like a train on a track, with the train
propelled by a powerful electrical pulse. Very high speeds have been
demonstrated in test models, and rail guns have been proposed as
an alternative to rockets for sending into outer space any object
that would be strong enough to survive the extreme accelerations.
Suppose that the rail gun capsule is launched straight up, and that
the force of air friction acting on it is given by F = be-cx, where x
is the altitude, b and c are constants, and e is the base of natural
logarithms. The exponential decay occurs because the atmosphere
gets thinner with increasing altitude. (In reality, the force would
probably drop off even faster than an exponential, because the capsule
would be slowing down somewhat.) Find the amount of kinetic
energy lost by the capsule due to air friction between when it is
launched and when it is completely beyond the atmosphere. (Gravity
is negligible, since the air friction force is much greater than the
gravitational force.) R
A certain binary star system consists of two stars with masses
m1 and m2, separated by a distance b. A comet, originally nearly at
rest in deep space, drops into the system and at a certain point in
time arrives at the midpoint between the two stars. For that moment
in time, find its velocity, v, symbolically in terms of b, m1, m2, and
fundamental constants. [Numerical check: For m1 = 1.5 × 1030 kg,
m2 = 3.0×1030 kg, and b = 2.0×1011 m you should find v = 7.7×104
An airplane flies in the positive direction along the x axis,
through crosswinds that exert a force F = (a + bx)◯ + (c + dx)ŷ.
Find the work done by the wind on the plane, and by the plane on
the wind, in traveling from the origin to position x.
In 1935, Yukawa proposed an early theory of the force that
held the neutrons and protons together in the nucleus. His equation
for the potential energy of two such particles, at a center-tocenter
distance r, was PE(r) = gr-1e-r/a, where g parametrizes the
strength of the interaction, e is the base of natural logarithms, and
a is about 10-15 m. Find the force between two nucleons that would
be consistent with this equation for the potential energy.
Prove that the dot product defined in section 3.7 is rotationally
invariant in the sense of book 1, section 7.5.
Fill in the details of the proof of A·B = AxBx+AyBy+AzBz
on page 65.
Does it make sense to say that work is conserved?
Solution, p. 160
(a) Suppose work is done in one-dimensional motion. What
happens to the work if you reverse the direction of the positive
coordinate axis? Base your answer directly on the definition of work.
(b) Now answer the question based on the W = Fd rule.
A microwave oven works by twisting molecules one way and
then the other, counterclockwise and then clockwise about their own
centers, millions of times a second. If you put an ice cube or a stick
of butter in a microwave, you'll observe that the oven doesn't heat
the solid very quickly, although eventually melting begins in one
small spot. Once a melted spot forms, it grows rapidly, while the
rest of the solid remains solid. In other words, it appears based on
this experiment that a microwave oven heats a liquid much more
rapidly than a solid. Explain why this should happen, based on the
atomic-level description of heat, solids, and liquids. (See, e.g., figure
b on page 35.)
Starting at a distance r from a planet of mass M, how fast
must an object be moving in order to have a hyperbolic orbit, i.e.,
one that never comes back to the planet? This velocity is called
the escape velocity. Interpreting the result, does it matter in what
direction the velocity is? Does it matter what mass the object has?
Does the object escape because it is moving too fast for gravity to
act on it? p