Special Relativity of Train Problem

• i_hate_math
In summary, the bomb in the rear car is set to deactivate when the front car passes the far end of the tunnel. However, the front car passes the near end of the tunnel first, so the bomb detonates.

Homework Statement

A relativistic train of proper length 237 m approaches a tunnel of the same proper length, at a relative speed of 0.951c. A paint bomb in the engine room is set to explode (and cover everyone with blue paint) when the front of the train passes the far end of the tunnel (event FF). However, when the rear car passes the near end of the tunnel (event RN), a device in that car is set to send a signal to the engine room to deactivate the bomb. Train view:(a) What is the tunnel length? ans: 77.3m (b) Which event occurs first, FF or RN? ans: FF (c) What is the time between those events? ans: 5.74E-7 s (d) Does the paint bomb explode? ans: YES Tunnel view:(e) What is the train length? ans: 77.3m (f) Which event occurs first? RN (g) What is the time between those events? ans: 5.74E-9 s (h) Does the paint bomb explode? ans: YESIf your answers to (d) and (h) differ, you need to explain the paradox, because either the engine room is covered with blue paint or not; you cannot have it both ways. If your answers are the same, you need to explain why.

L=L0*sqrt(1-ß^2)

The Attempt at a Solution

I got part a thru g correct, what I could not understand is the answer to part (h) is yes. I believe from the tunnel's point of view the train (L=77.3m) has its rear passes thru the near end of tunnel (L0=273m) before its front passes thru the far end. This indicates that the bomb is deactivated first.

i_hate_math said:
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(h) Does the paint bomb explode? ans: YES
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I believe from the tunnel's point of view the train (L=77.3m) has its rear passes thru the near end of tunnel (L0=273m) before its front passes thru the far end. This indicates that the bomb is deactivated first.
Well then, why did you say that the paint bomb explodes in this case?

SammyS said:
Well then, why did you say that the paint bomb explodes in this case?
Because the correct answer is the bomb will go off

i_hate_math said:
Train view:(a) What is the tunnel length? ans: 77.3m
I don't agree with this answer. Perhaps it's just a typographical error?
I believe from the tunnel's point of view the train (L=77.3m) has its rear passes thru the near end of tunnel (L0=273m) before its front passes thru the far end.
Yes
This indicates that the bomb is deactivated first.
It's not that simple. Don't overlook this part of the problem:
"However, when the rear car passes the near end of the tunnel (event RN), a device in that car is set to send a signal to the engine room to deactivate the bomb."

i_hate_math
i_hate_math said:

Homework Statement

A relativistic train of proper length 237 m approaches a tunnel of the same proper length, at a relative speed of 0.951c. A paint bomb in the engine room is set to explode (and cover everyone with blue paint) when the front of the train passes the far end of the tunnel (event FF). However, when the rear car passes the near end of the tunnel (event RN), a device in that car is set to send a signal to the engine room to deactivate the bomb. Train view:(a) What is the tunnel length? ans: 77.3m (b) Which event occurs first, FF or RN? ans: FF (c) What is the time between those events? ans: 5.74E-7 s (d) Does the paint bomb explode? ans: YES Tunnel view:(e) What is the train length? ans: 77.3m (f) Which event occurs first? RN (g) What is the time between those events? ans: 5.74E-9 s (h) Does the paint bomb explode? ans: YESIf your answers to (d) and (h) differ, you need to explain the paradox, because either the engine room is covered with blue paint or not; you cannot have it both ways. If your answers are the same, you need to explain why.

L=L0*sqrt(1-ß^2)

The Attempt at a Solution

I got part a thru g correct, what I could not understand is the answer to part (h) is yes. I believe from the tunnel's point of view the train (L=77.3m) has its rear passes thru the near end of tunnel (L0=273m) before its front passes thru the far end. This indicates that the bomb is deactivated first.
i_hate_math said:
Because the correct answer is the bomb will go off
Oh! So those weren't your answers for a - h, they were the ' correct ' answers.

What is the location of the bomb activation/deactivation system?

TSny said:
I don't agree with this answer. Perhaps it's just a typographical error?
Yes
It's not that simple. Don't overlook this part of the problem:
"However, when the rear car passes the near end of the tunnel (event RN), a device in that car is set to send a signal to the engine room to deactivate the bomb."
Yes you're right about the contracted length, a typo it is indeed. The correct value should be 73.3m (3sig figs)

Thanks a lot for highlighting the line where it says the "device sends a signal". I see that the signal being sent out does not assure that it reaches the front of the train in time.

SammyS said:
Oh! So those weren't your answers for a - h, they were the ' correct ' answers.

What is the location of the bomb activation/deactivation system?
the bomb is located at the front of the train, while there is another device, located at the rear, which can send out a signal to deactivate the bomb. also I mistyped the answer to part a and e, the correct answer should be 73.3m

It's not clear as to what point the writer of the question intended to make. The entire train will be well out of the tunnel before the deactivation signal can reach the locomotive. This is true even if the deactivation signal is sent at the moment (in the train's frame) that the locomotive first enters the tunnel.

Suppose we alter the problem in the following way, so that the relative delay in the deactivation signal is taken out of play.
Let's place the blue paint bomb at the center of the train. A sensor at the front of the train sends a signal (perhaps a flash of light) to detonate the paint bomb as the front of the train passes the far end of the tunnel. Similarly, a sensor at the rear of the train sends a signal to deactivate the paint bomb as the the rear car passes the near end of the tunnel.​

Under this situation, the answers are the same as for the original situation (with the 73.3m corrected value). How do you explain answer h in this case?

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i_hate_math said:
Thanks a lot for highlighting the line where it says the "device sends a signal". I see that the signal being sent out does not assure that it reaches the front of the train in time.
Can you show that it is impossible for the signal to reach the front of the train before the bomb goes off?