Ball dropped from a moving train

In the reference frame of the platform the ball has initial horizontal velocity equal to the velocity of the train. The vertical direction is the same observed on the train because both observers agree that gravity is acting on the ball causing an acceleration -g. But, in the platform's reference frame it already had x-velocity, and the resultant path is a parabola.

 

I am currently reading about this in Einstein's writings. When the person on the train drops the ball and looks down he just sees it fall straight down because it is moving at the same velocity he is.

However, a person standing on a platform (not in motion) sees the ball fall down and perpendicular to his view. (In two dimensional motion)
It is all about the reference point. Relativity is a very cool and interesting topic!

 

Just to add a little bit: the reason it is a parabolic path is because the motion parallel to the direction of the train's motion is constant (equal to the speed of the train); meanwhile the position in the perpendicular direction (straight down) goes as t^2. It's this time squared that gives it a quadratic (parabola) shape.