You are travelling in a train that is slowing down upon approaching the station. You throw a heavy ball, aiming it directly at the ceiling above your chair. Relative to you, where will the ball fall? Explain your answer.
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The ball will fall directly from the position you threw it off. From your position you are not aware if the train is moving or has stopped. Because you have no knowledge then from your perspective you see the ball fall directly where it left.
I don't know exactly what level of relativity you are taking but I derived my conclusion from basic knowledge of relativity.
The train is slowing down. As such, it is decelerating. Once you have thrown the ball up, it will maintain its original velocity due to inertia. Therefore, it will accelerate forward relative to you, and will eventually fall to the ground ahead of you.
(Special relativity applies to *inertial* reference frames only; the train is a non-inertial reference frame because it is decelerating).
The ball will fall definitely ahead of you. This is because when you've thrown the ball it will gain the forward velocity equal to velocity of you at that instant and that is equal to velocity of the train. Neglecting air resistance we can calculate the distance traveled by the ball in forward direction in the time which it spends in air. Now this is definitely more than the distance travel by you or train because train velocity is decreasing with time but unless the ball touches the ground it's forward velocity remain almost constant equal to the initial velocity with which you have thrown the ball.