A ball is thrown from rear end of the compartment of train to the front end whic

Observer A is in the train's frame, which is inertial (constant velocity). For both observers, the vertical motion is determined solely by the initial vertical velocity and gravity, which are identical because the train's velocity has no vertical component. Thus, the time of flight and vertical range are the same. Horizontally, observer A sees the ball's horizontal velocity relative to the train, while observer B sees the sum of that and the train's velocity. Since time of flight is equal, the horizontal ranges differ: observer B sees a larger horizontal range (if the train moves forward). Hence, vertical ranges are equal, horizontal ranges differ.