Problem Set #02 (Kinematics in Special Relativity)

Physics 322: Modern Physics
Spring 2015
Problem Set #02 (Kinematics in Special Relativity)
Due Friday, January 30 at 12 noon (in lecture)
ASSUMED READING: Before starting this homework, you should read
Chapter 1 and Chapter 2.6 through 2.10 (you can skim the sections on General
Relativity) of Harris’ Modern Physics.
1. [Harris 2.22 modified] According to Bob on Earth, Planet Y
(uninhabited) is 5 ly away. Anna is in a spaceship moving away from
Earth at 0.8c. She is bound for Planet Y to study its geology.
Unfortunately, Planet Y explodes. Bob sees the explosion 7 years after
Anna left Earth, so knowing Planet Y is 5 ly away, works out the time of the
explosion to be 2 years after Anna passed Earth. Call the passing of Anna
and Bob time zero for both.
a. According to Anna, how far away is Planet Y when it explodes?
b. At what time does it explode?
c. What do these results mean? In other words, explain what is
actually going on here.
2. [Harris 2.25 (tweaked)] Anna and Bob are in
identical spaceships, each 100 m long. The
diagram shows Bob’s view as Anna’s ship passes
at 0.8c. Just as the backs of the ships pass one
another, both clocks read 0. At the instant
shown, Bob Jr., on board Bob’s ship, is aligned
with the front of Anna’s ship. He peers through a
window in Anna’s ship and looks at the clock.
a. In relation to his own ship, where is Bob
Jr.? Explain your reasoning.
b. What does the clock he sees read? Explain your reasoning.
3. [Harris 2.53 modified] At rest, a light source emits 532 nm light.
a. As it moves along the line connecting it and Earth, observers on
Earth see 412 nm. What is the source’s velocity (magnitude and
direction)?
b. Were it to move in the opposite direction at the same speed, what
wavelength would be seen?
c. Were it to circle Earth at the same speed, what wavelength would be
seen?
d. One (incorrect) model we give introductory students for Doppler
Shift is that the waves get “crunched up” or “stretched out” between
the source and the observer moving relative to each other, thus
causing a change in wavelength. If an object is circling the Earth,
you might expect it to exhibit no Doppler Shift, since it is neither
moving towards or away from us. Why did you see a Doppler shift
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Physics 322: Modern Physics
Spring 2015
in part (c) [and now you know you
should have!]? HINT: “Time
dilation” is the answer, the hard part
is grasping why!
4. [Variant of Harris 2.12] Can two moving objects of mass 2 and 3 (in
arbitrary units) collide and as a consequence form a single object of mass
less than 5 (in the same arbitrary units) with no additional debris?
Explain your reasoning.
5. Assuming for a moment that the relativistically correct expression for
momentum is the “correct” expression, determine at what speed the
classical expression for the momentum is significantly incorrect (let’s say a
10% error). HINT: The ratio of the relativistic expression for momentum
to the classical expression would differ by 10% from 1.0 at this point.
6. [Harris 2.62 (tweaked)] In a particle collider experiment, particle 1 is
moving to the right at 0.99000c and particle 2 is moving to the left at
0.99000c, both relative to the laboratory. What is the relative velocity of
the two particles according to (an
observing moving with) particle 2?
7. [Harris 2.76] In the collision shown,
energy is conserved, because both objects
have the same speed and mass after as
before the collision (since this copy of the
image may be unclear, the particle on the
left has a mass of 16 before and after the collision, the particle on the right
has a mass of 9 before and after the collision). Since the collision merely
reverses the velocities, the final (total) momentum is opposite the initial.
Thus, momentum can be conserved only if it is zero.
a. Using the relativistically correct expression for momentum, show
that the total momentum is zero – that is that momentum is
conserved. (Masses are in arbitrary units).
b. Using the relativistic velocity transformation, find the four
velocities in a frame moving to the right at 0.6c. One of the
velocities should be easily predictable. Is it correct?
c. Verify that momentum is conserved in the new frame.
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