A spaceship of proper length L_p = 400 m moves past a transmitting station at a

Question

A spaceship of proper length L_p = 400 m moves past a transmitting station at a speed of 0.85c. (The transmitting station broadcasts signals that travel at the speed of light.) A clock is attached to the nose of the spaceship and a second clock is attached to the transmitting station. The instant that the nose of the spaceship passes the transmitter, the clock attached to the transmitter and the clock attached to the nose of the spaceship are set equal to zero. The instant that the tail of the spaceship passes the transmitter a signal is sent by the transmitter that is subsequently detected by a receiver in the nose of the spaceship. When, according to the clock attached to the nose of the spaceship, is the signal sent? When, according to the clocks attached to the nose of the spaceship, is the signal received? When, according to the clock attached to the transmitter, is the signal received by the spaceship? According to an observer that works at the transmitting station, how far from the transmitter is the nose of the spaceship when the signal is received?

Explanation / Answer

(a)   According to the clock attached to the nose of the ship, is the signal sent       time tA   =   L0 / v    = 300 m / ( 0.65*3*108 m/s)   = 1.53 s (b)   according to the clock attached to the transmitter, is the signal received by the spaceship          t   =   time taken to travell length of the ship +    tA                   = (    L0 / c ) + 1.53 s   =    1.00 s +   1.53 s      =   2.53 s          t   =   time taken to travell length of the ship +    tA                   = (    L0 / c ) + 1.53 s   =    1.00 s +   1.53 s      =   2.53 s (c) According to inverse transformation    tB ' = ( t - vx /c2 )    ...............(1) Here    =   1/ sqrt ( 1 -v2 /c2 )   Here v = - 0.65 c ,   t =   2.53 s ,   x = L0   = 300 m =   1.31 plug all values in equation (1) we get      tB '    = 4.165 s      tB '    = 4.165 s (d)    x'   =   ( x - vt ) Here     t =   2.53 s , = 1.31 ,   v =  - 0.65*3*108 m/s     x'   = 1039.2885    =   1.039 km         

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