Question

A spaceship of proper length L_p = 500 m moves past a transmitting station at a speed of 0.83c. (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.

(a) When, according to the clock attached to the nose of the spaceship, is the signal sent?
  µs

(b) When, according to the clocks attached to the nose of the spaceship, is the signal received?
  µs

(c) When, according to the clock attached to the transmitter, is the signal received by the spaceship?
  µs

(d) 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?
m

Answer 1

Let us first visualize the problem :

let S be the reference frame of the spaceship and S^' be that of he transmitting station. Let event be the emission of the light pulse and event B the reception of the light pulse at the nose of the spaceships.

In (a) and (c) sub-part of the question we can use the classical distance, rate, and time relationship and in the (b) and (d) we can apply the inverse Lorentz transformations.

(a) In both S and S^' the pulse travels at the speed c. Thus

  

(c) The elapsed time, according to the clock of the ship is:

To find the time of travel of the pulse to the nose of the ship:

  

Substitute numerical values and evaluate t_B :

(b) The inverse time transformation is:

  

Substitute the numerical values and evaluate t_B^' :

  

(d) The inverse transformation for x^' is :

Substitute numerical values and evaluate x^':