Question

A relativistic train of proper length 249 m approaches a tunnel of the same proper length, at a relative speed of 0.922c. 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? (b) Which event occurs first, FF or RN? (c) What is the time between those events? (d) Does the paint bomb explode? Tunnel view: (e) What is the train length? (f) Which event occurs first? (g) What is the time between those events? (h) Does the paint bomb explode? If 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. (a) Number Number Units (b) (c) Number Units (d) (e) Number Units (f) (g) Number Units 4 (h)

Answer 1

a) tunnel length as seen by train is conctracted

v2 L = LO L011 2

L0 is proper length

so we get

$L=96.41m$

b) as view by the tarin length of tunnel is shorter than train so front of train passes far end of tunnel so FF occurs first

c) time diference is

$\Delta t=\frac{L_0/v}{\sqrt{1-\frac{v^2}{c^2}}}$

so we get

$\Delta t= 6.62 \times 10^{-6}s= 6.62 \mu s$

d)now because FF event occur first so bomb will explode

e)so due to relative speed of train length will contract and

v2 L = LO L011 2

L= 96.41 m

f) now the RN event occur first

g) so time between those events is

$\Delta t=\frac{L}{v}=3.48\times 10^{-7}s= 0.348\mu s$

h)now because as from tunnel view RN event occur first so bomb will not explode

There is parodox in both events as seen by tunnel and train view but it is actually not possible to decide either it explode or not because above argument is made over view by either train or tunnel but physically both of there length is same so both event occur simultaneously