Question

Mary is sitting exactly in the middle of a railway carriage as it travels along a straight track. There are mirrors on the inside end walls of the carriage: mirror A is at the front of the carriage and mirror B at the back. Mary measures both mirrors to be a distance d0 = 30 meters away from her, and the length of the carriage to be 60 meters.
Peter is a train spotter, who is standing by the side of the track at a railway station, observes that the through train is travelling extremely fast, at v = c/2, where c is the speed of light. He also notes that, at the very instant Mary passes him, she starts to light a match. This question refers to the first spark from this match. Both Mary and Peter have set their watches so that this event corresponds to time zero. Some light from the spark initially travels in the direction of the train’s motion, is reflected from mirror A and returns to Mary; other light from the same spark initially travels in the opposite direction, is reflected from mirror B and also returns to Mary. You are required to answer the following parts (a) to (e).

(a) According to Mary, find the following time,
(i)   when light from the spark reaches mirror A;
(ii) when light from the spark reaches mirror B;
(iii) when light from the spark returns to Mary from mirror A; and
(iv) when light from the spark returns to Mary from mirror B.

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(b) Similarly, find the times according to Peter, when the same four events take place. One way of answering this question is to start from Mary’s space and time coordinates of the four events and then apply the Lorentz transformation equations. In applying these equations note that it is permissible to choose Mary’s frame on the moving train as the unprimed frame and Peter’s frame on the ground as the primed frame. (This choice has some advantages in simplifying the algebra, but you will need to take care in allocating signs to physical quantities.)

Use your answers in parts (a) and (b) to answer the following:

(c) Do Peter and Mary agree that light from the spark reaches mirror A at the same time as mirror B? Do they agree that the light returning from mirror A reaches Mary at the same time as the light returning from mirror B? Comment on your answers, drawing out general principles about the relativity of simultaneity in special relativity.

(d) Do Peter and Mary agree that the time taken for light to travel from the spark to mirror A is the same as the time taken for light to travel from mirror B to Mary? Give a simple argument explaining why such agreement or disagreement is reasonable.  

(e) Use two of the times calculated in parts (a) and (b) to illustrate what is meant by time dilation in special relativity.

Answer 1

PART A

1)

According to Mary

SINCE SPEED OF LIGHT IS INVARIANT IN ANY INERTIAL FRAME

Distance of mirror A from Mary = 30 m, Velocity of light = c

Time = 30/c =10^-7 sec.

ii)

Distance of mirror B from Mary Is same = 30 m, Velocity of light = c

Time = 30/c =10^-7 sec.

iii)

Total distance covered in returning to Mary, 2*30 = 60m, Velocity = c

Time = 60/c = 2*10^-7 sec

iv)

Distance 2*30 = 60, Velociy =c

Time = 2*10^-7 sec

Part B

All time measured in part a are proper time as it measured in the rest frame of Mary.

For Peter Train is moving with velocity v = c/2

we can take all the measurement of time in Mary frame as Event.

Now From the consequence of relativity, (Time dialation) we can calculate time of all the event as measured in frame S(Peter frame)

i)

1/(1 - V²/C²) = 2/3 = 1.155

t' = t = 1.155*10^-7 sec.

ii)

t' = 10155*10^-7 sec

iii)

t' = 1.155*2*10^-7 sec = 2.31 * 10^-7 sec.

iv)

t' = 2.31 * 10^-7 sec.

PART C

YES, Mary and Peter seperataly agree that spark reaches mirror A and mirror B at the same time.

YES, they both seperataly agree that light returning from mirror A reaches Mary at the same time as the light returning from mirror B.

Comment : From the postulate of special theory of relativity we know that speed of light is invariant ( has same value ) in all the inertial frame, So, Time taken by light to caver same distance will be same. Thus the 1st two event will be simultaneous and the next two events will be simultaneous.

PART D

Since time measured by Mart will be proper time and time measured by Peter will be dialated time Both time will be different, Time measured by peter will be more than time measured by Mary because of consequence of relativity.

But For Peter and Mary seperataly Time measured will be same as it is measured in their reference frame for both the event.

PART E

Since time calculate in part A is the proper time which is the time measured in the rest frame of obeserver where as time measured in part B is dialated time So time measured in part B is more than time measured in part A.

We can come to conclusion that clock ticks slowerr in moving frame than in stationary frame.And this dialation of time is known as time dialation in special relativity.