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Throwing a stone from a moving train involves the same type of calculations as throwing a stone from a stationary platform. The difference is that you now consider the added horizontal velocity imparted by the moving train. If you throw forward, the train's velocity is added to your contribution to the stone. If you throw backward, the train's velocity is subtracted from your contribution to the stone. If you throw sideways, the train's velocity does not alter your contribution to the stone. Whether any of this matters or not depends on friction due to the different air velocities encountered by the stone. Vertical velocity is relatively unchanged by the train's contribution. The stone will still go up (if you include an upward vector in your throw) and it will still go down. The end result is that the stone will hit the ground at some point. That point will be relatively the same in both cases, except for the minor difference due to air velocity.

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Q: What will happen when a stone is thrown from a moving train?
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