Answer to Question #98256 in Mechanics | Relativity for AbdulRehman

Question

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.
iii) Describe the events shown on your diagrams from the point of view
of the robbers (either one). Be sure to state the order in which the
events occur in their frame of reference, to discuss whether the train
can escape, and to explain why or why not.

Answer 1

From the point of view of the robbers they blow up the exit of the tunnel when the front of the train enters the entrance to the tunnel. Assume that the vertical distance between the train roof and the stones right above the roof (i.e. height of the tunnel minus height of the train) is $y=0.6$ m. The time required for the stones to reach the roof is

$$t_v=\sqrt{2y/g}=\sqrt{2\cdot0.6/9.8}=0.35\text{ s}.$$

How long for such a fast train does it take to go through the tunnel in the frame of reference of the robbers?

$$t_h=2L/v_{\text{train}}=\\=2\cdot100/(0.8\cdot3\cdot10^8)=8.33\cdot10^{-7}\text{ s}.$$

At such a speed more than

$$N=t_v/t_h=419911$$

trains can without any trouble go through the tunnel.