# Relativistic train problem

#### jianxu

1. The problem statement, all variables and given/known data
A train moves past a tower at a speed of 0.78c. The train has a rest length of L' = 450m. When the front of the train passes the tower, the clocks in the tower and in the front of the train are synchronized to t=t'=0s. The instant that the back of the train passes the tower, the tower sends a signal which is subsequently detected by the receiver in the front of the train.

a) according to the clock on the train, when is the signal sent?
b) when according to the clock in the tower, is the signal received at the front of the train
c)when according to the clock on the train is the signal received?
d) according to an observer on the tower, how far away is the front of the train when the signal is received?

2. Relevant equations

Time dilation equations
Length Contraction
lorentz transformation

3. The attempt at a solution
for part a, I simply used length/speed which gave me the time.
here I used 450m/0.78c = 1.92x10^-6 s.

The biggest problem I have is part b and c.
I was thinking of applying lorentz transformation:
t=$$\gamma$$(t'+((vx')/c^2))

where the change in x is the difference between the front of the train to the tower. This should give me the time the signal is sent in the tower's frame.

Also, I applied length contraction to the train to obtain the change in x in the tower's frame. Ultimately though, I don't know how to put all the ideas together.

For part d, I think the only equation that's needed is:
change in x = v * change in t where change in t is the time between signal received and signal sent and x is the distance traveled during time t plus the initial length(after length contraction).

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#### jianxu

for part b and c, can I set the x position of the train signal and tower signal equal each other and then solve for the time of when they intersect?

#### jianxu

I think part c can be achieved by applying time dilation but I still don't understand part b. I applied length contraction to the train but I am stuck after that. Please help!!! :(

#### jianxu

My professor said we can assume the signal sent travels at a speed of light. I'm still stuck, if anyone know how to solve part b, or verify if my thoughts for the other parts are correct, please let me know. Thanks.

#### granpa

for part b and c, can I set the x position of the train signal and tower signal equal each other and then solve for the time of when they intersect?
train signal?

#### Doc Al

Mentor
To solve part b, answer these questions:
(1) According to the tower clock, when is the signal sent?
(2) According to the tower clock, how long does it take for the signal to travel to the front of the train? (Set up a kinematic equation to solve for this time.)

#### jianxu

To solve part b, answer these questions:
(1) According to the tower clock, when is the signal sent?
(2) According to the tower clock, how long does it take for the signal to travel to the front of the train? (Set up a kinematic equation to solve for this time.)

well for part 1 I got 1.19X10^-6 by using distance(after length contraction) divided by speed = time

as for part 2, can I say that :
for the tower x = vt
and for the train, x = x0 + vt
x0 is the initial length contracted distance between the train and the tower?

I was also wondering if I were to approach the other parts as I had described, would they be correct? Thanks

#### Doc Al

Mentor
well for part 1 I got 1.19X10^-6 by using distance(after length contraction) divided by speed = time
Looks good.
as for part 2, can I say that :
for the tower x = vt
Careful: Realize that the distance traveled by the signal travels at speed c, so x = ct.
and for the train, x = x0 + vt
x0 is the initial length contracted distance between the train and the tower?
What is that initial distance between the front of the train and the tower?
I was also wondering if I were to approach the other parts as I had described, would they be correct?
Yes, if I understand what you're saying.

#### jianxu

Ah ok, I was so busy thinking about just the relativity stuff I didn't even think about using kinematics for this problem. Thank you very much for reminding me about the kinematic portion. Have to remember to apply previous knowledge next time. Thanks again Doc!!!!!

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