A ball is thrown straight up by a passenger in a train that is moving with a constant velocity. Where would the ball land-back in his hands, in front of him, or behind him? Does your answer change if the train is accelerating in the forward direction? If yes, how?

When the train is moving with a constant velocity, both the passenger and the ball share the same horizontal velocity. Since the ball is thrown straight up, it will experience only gravitational force acting on it vertically. No additional horizontal forces will be acting on the ball. As a result, the horizontal velocity of the ball remains constant throughout its ascent and descent. Therefore, the ball will continue to move horizontally in sync with the passenger, and the ball will land back in the passenger's hands.

When the train accelerates in the forward direction, the passenger and the ball will experience an additional force due to acceleration. At the moment when the ball is thrown straight up, the ball will have the same initial horizontal velocity as that of the passenger. However, the train continues to accelerate, while the ball does not experience any horizontal force after leaving the passenger's hand. Due to the absence of horizontal force acting on the ball, the ball's horizontal velocity will remain constant throughout its flight. In the meantime, the train and the passenger continue to accelerate. As they accelerate forward, the passenger gains horizontal velocity relative to the ball. Consequently, the ball will lag behind the passenger's moving position and will land behind the passenger.