Question

1. Alice is moving wit 0.8c towards a mirror. She sends a light pulse towards the mirror when she is at a distance L from the mirror. Bob is at rest with respect to the mirror and is observing this experiment. After how long will Alice receive the reflected pulse, as measured by Alice and Bob?

Answer 1

The second postulate of special theory of relativity is that speed of light is constant, it doesn't depend on the velocity of the frame of reference. In our problem Alice is moving with a velocity 0.8C and she send a light pulses towards the mirror. No matter how fast Alice moves she will always see that light moves with speed C towards the mirror.

Same thing happens for Bob, no matter whether Bob is at rest or moving he will always observe that speed of light is constant and equal to C.

Let after a time interval t Alice will receive the reflected light pulse in her frame which is moving with a constant velocity of 0.8C and in this time his frame moves a distance X towards the mirror. So in time t frame of Alice moves a distance X with a speed 0.8C. So we can write mathematically

Similarly light pulses travels a distance L towards the mirror and after reflection it travels (L-X) distance towards Alice in time t. So we can write

Now eliminating the value of X from the above two equation we get

This the time after which Alice will receive the reflected pulse

Now in Bob frame of reference he will see that Alice is moving with a speed of 0.8C towards the mirror. So according to the special theory of relativity there will be time dilation. Let the time measured by Bob in his frame is T then the value of T is given by

Where is the velocity of the frame of Alice which is 0.8C. so the value of T is

This is time measured by Bob after which Alice will receive the reflected light pulse.