Answers to Exercises
The engines on a 4000-kg jet plane as_celerating for
take-off exert a constant tluust of magnitude I fu, I = 20,000 N
on the jet as it accelerates from rest a distance D = 1000 meters
down the runway before taking off. This take-off is observed
by someone riding in a train with a constant speed of 30 m/s
alongside the runway. Assume that the train and jet move
in the same direction relative to the ground (which we will
take to be the +x direction), and assume that both the pas­
senger and the plane are at x = 0 in the ground frame when
the plane starts its run at t = 0.
(a) In unit C, we saw that if the net external force foe, on
an object is constant, the change in the object's kinetic
en�rgy during a given interval of time should be equal
to fnet • �rCM, where �rcM is the displacement of the
object's center of mass during that time. Use this to
show that the jet's speed relative to the ground at take­
off is 100 m/s. Explain your reasoning.
(b) Assuming that the jet's acceleration is constant, show
that it takes it 20 s to reach this speed. Explain.
(c) What are the plane's initial and final x-velocities in the
train frame? Explain.
(d) Assume that the passenger's position defines the ori­
gin x' = 0 in the train's frame. What is the jet's initial
x-position in the train frame?
(e) What is its x-position at take-off in the train frame?
Explain.
(f) Show that the change illthe jet's kinetic energy K' in the
train frame is equal to f�et • �rcM in that frame. (Thus,
this law of physics, the momentum requirement, is the
same in both frames, even though the numerical values
of K = Fnet . �;:CM and K' = f�et . �;:CM are not.)
RlM.9
I
r
RlM.10 A person in an elevator drops a ball of mass m from
rest from a height h above the �evator floor. The elevator
is moving at a constant speed I /31 downward with respect
to its enclosing building.
(a) How far will the ball fall in the building frame before it
hits the floor? (Hint: >h!)
(b) What is the ball's initial vertical velocity in the build­
ing frame? (Hint: Not O!)
(c) Use the law of conservation of energy in the building
-frame to compute the ball's final speed (as measured
in that frame) just before it hits the elevator floor.
(d) Use the Galilean velocity transformation equations
and the result of part (c) to find the ball's final speed in
the elevator frame.
(e) Assume that conservation of energy applies in the
building's frame. Use the result of part (d) and the fact
that the ball's acceleration is I g I to show that energy is
also conserved in the elevator frame.
f'
21
Derivation
R1 D.1 A totally symmetric way to orient a pair of reference
frames is so that their +x directions point in the direction
that the other frame is moving. How is this different from
the "standard" orientation (draw a picture)? How would the
Galilean position and velocity transformation equations be
different if we were to use this convention?
R1D.2 Imagine two inertial reference frames in standard
orientation, where the Other Frame moves in the +x direc­
tion with x-velocity /3 relative to the Home Frame. Suppose
an observer in the Home Frame observes the following col­
lision: an object with mass' m 1 and velocity v1 hits an object
with mass 1112 traveling with velocity v2• After the collision,
the objects move off with velocities 3 and 4, respectively.
Do not assume that all or even any of these velocities are in
the x direction. Assume, though, that total momentum is
measured to be conserved in the Home Frame, that is, that
v
m1v1 + m2v2 = m1v3 + mi'i\
v
(assume this!)
(Rl.7)
Using this equation and the Galilean transformation equa­
tions, show that if the Newtonian view of time is correct,
then the total momentum of the two objects will also be
conserved in the Other Frame
m1v; + m2v; = m1v; + m2v�
(prove this!)
(Rl.8)
even though the velocities measured in the two frames are
very different. Please show your work in detail.
Rich-Context
R1R.1 Design a first-law detector that does not use a float­
ing ball as the basic active element. Your detector should
primarily test Newton's first law and not some other law
of physics (although it is fine if other laws of physics are
involved in addition to the first law). Preferably, your
detector should be reasonably practical and (if at all pos­
sible) usable in a gravitational field. (Note: There are many
possible solutions to this problem. Be creative!)
RlR.2 Consider the economy reference frame described in
problem RlM.2. Prove that an object that actually moves at
a constanltvelocity close to that of light will be observed in
the economy frame to move faster as it approaches the ori­
gin and slower as it departs. Also describe what happens if
the object movesfaster than light.
ANSWERS TO EXERCISES
RlX.1 Imagine that the Home Frame is an inertial frame.
Consider a set of isolated objects arrayed around the Home
Frame that happen to be initially at rest in that frame. Since
their velocity with respect to the inertial Home Frame has
to be constant, tnese objects will remain at rest relative to
the Home Frame. Now, if the Home Frame moves at a con­
stant velocity relative to the Other Frame (and the latter is
rigid and nonrotating so that all parts of the Other Frame
move with a constant velocity relative to the Home Frame,
and vice versa), then our set of isolated objects must also
move with a constant velocity relative to the Other Frame.
Since these isolated objects are observed to move with a
constant velocity everywhere in the Other Frame, it must
be an inertial frame as well.