If a train of length l is stationary, then light from a bulb flashed at the midp

Question

If a train of length l is stationary, then light from a bulb flashed at the midpoint of the train will reach the two ends at the same time. If the train moves past a station platform at speed v, the (in the frame of the station) light from a single flash of this bulb will no longer reach the two ends of the train at the same time, since the rear end of the train is moving toward the light while the front end is moving away. Evidently if we want to place the bulb so the light reaches the ends of the train simultaneously in the frame of the station, we must move the bulb closer to the front of the train; i. e., we must attach it at a point a fraction x of the total length of the train behind the front, where x is less than 1/2. Show that x = l/2 (1 - v/c)

Explanation / Answer

Let train velocity is v and velocity of light is c,

from reference point ( station)

front and, velocity is = c-v

rear end velocity= c+v

let us take if bulb is at distance x from front and from rear distance = L-x for same time

so, applying time = distance/velocity

so, x/(c-v) = (L-x)/(c+v)

x(c+v)=(L-x)(c-v)

xc+xv=cL-vL-xc+xv

xc+xv+xc-xv=cL-vL

x= L(c-v)/2c

x = (L/2)(1-v/c)

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