Please ignore all relativity stuff.
Consider two cases.
Case 1: Train moving at constant speed.
Case 2: Train with constant acceleration.
I'm standing on the ground with a ball in my hand. I throw it towards a moving train door.
I know everything inside the train should travel at train speed.
Now, when I throw the ball I know it has certain velocity traveling away from me.
When the ball enters the train door will it acquire the velocity of the moving train in the direction of train?
Will the ball pass through the other door?