Question

1. An observer (O) is standing on a long tunnel that has a rest length of 12 c^.µs.  O sees a long train that he measures to be 12c^.µstravel through the tunnel at a speed of 0.600c. The front and back of the tunnel are mounted with strobe lights so that when the front or end of the train reaches the end of the tunnel the light flashes.  So we have 4 events (flashes).

Event 1:  Front of train enters tunnel.

Event 2:  Front of train exits tunnel.

Event 3:  End of train enters tunnel.

Event 4:  End of train exits tunnel.

1. With the first event being (x=0, t=0) list the xand tcoordinates of each event in O’s IRF.
2. For an observer moving with the train (),what is the length of the train and what is the length of the tunnel?
3. With the first event being (x’=0, t’=0) use the Lorentz transformation to list the x’and t’ coordinates of each event in ’s IRF.
4. Are there two events in the tunnel’s IRF that are simultaneous?  Are these events simultaneous in the train’s IRF?  If not, what is the time elapsed between these two events?
5. Does the train fit in the tunnel?
6. Make a graph of the motion of both ends of the train and both ends of the tunnel in (x,t) and in (x’,t’).  (Put t on the vertical axis and x on the horizontal axis.

Answer 1

a.

Event 1: (0,0)

Event 2: Time taken by train to exit the tunnel will be

so, coordinates will be(12 c µs,20µs)

Event 3

Since the length of the train and tunnel is same in O's IRF. so, these two event occur at the same time but different x.

coordinates(0,20µs)

Event 4

It will take additional 20µs for back of the train to exit the tunnel. so, coordinates are (12 c^.µs, 40µs).

b) Original length of the train:

length in O's frame =Original length/gamma

Now, in train's frame the track is moving so length of track

c) Lorentz transfornations are

Event 1: (0,0)

Event 2 (x,t)=(12 c µs,20µs)

Event 3: (x,t)= (0,20µs)

Event 4 (x,t) (12 c^.µs, 40µs)

d) Two events simultaneous in tunnel's frame are event 2 and event 3 but these events are not simultaneous in train's frame. time elapsed=6µs

e) It depends upon the frame, for tunnel's frame it fits. For, trains frame it does not when everything is in rest train does not fit inside the tunnel.