Notes on Relativity and Cosmology - Physics Department, UCSB

3.8. HOMEWORK PROBLEMS 79
once each second, how much time passes in your reference frame between
two of it’s ticks?
Hint: Your calculator may not be able to deal effectively with the tiny
numbers involved. As a result, if you just try to type formula (3.3) into
your calculator you may get the value 1. What I want to know is how
much the actual value is different from 1. In cases like this, it is helpful to
use Taylor series expansions. The expansions you need to complete this
problem were given in problem (2-2).
3-7. A muon is a particle that has a lifetime of about 10 −6 seconds. In other
words, when you make a muon, it lives for that long (as measured in its
own rest frame) and then decays (disintegrates) into other particles.
a) The atmosphere is about 30km tall. If a muon is created in the upper
atmosphere moving (straight down) at 1 2c, will it live long enough to
reach the ground?
b) Suppose that a muon is created at the top of the atmosphere moving
straight down at .999999c. Suppose that you want to catch this muon
at the surface and shoot it back up at .999999c so that it decays just
when it reaches the top of the atmosphere. How long should you hold
onto the muon?
3-8. For this problem, consider three inertial observers: You, Alice, and Bob.
All three of you meet at one event where your watches all read zero. Alice
recedes from you at 1 2 c on your left and Bob recedes from you at 1 2 c on
your right. Draw this situation on a spacetime diagram in your frame of
reference. Also draw in a light cone from the event where you all meet.
Recall that you measure distances using your own lines of simultaneity,
and note that you find each of the others to be ‘halfway between you and
the light cone’ along any of your lines of simultaneity.
Now, use the above observation to draw this situation on a spacetime
diagram in Alice’s frame of reference. Use this second diagram to estimate
the speed at which Alice finds Bob to be receding from her. What happens
if you draw in another observer (again meeting all of you at t = 0) and
traveling away from Bob on the right at 1 2c as measured in Bob’s frame of
reference??
3-9. Consider a specific version of the ‘train paradox’ that we discussed in class.
Suppose that we a have tunnel of length 100m (measured in its rest frame)
and a train whose length is 100m (measured in its rest frame). The train
is moving along the track at .8c, and the robbers want to trap it using the
following scheme: one robber will sit at the entrance to the tunnel and
blow it up just after the back end of the train has entered, while the other
robber will sit at the exit of the tunnel and blow it up just before the front
end of the train gets there.