I may be misinterpreting the question, (as others have done previously), but from what I gather.... this high speed ball bouncer is only causing the ball to travel in a vertical direction in the rest frame of the train. The train is moving perpendicular to that motion. The path that the ball in the moving train travels (the zig-zag-esque path) is greater than the path of the ball moving at rest (straight up and down). The velocity of the ball on the train is also faster than that of the ball on the ground... since the ball on the train has some x-component (ball speed equal to that of the train speed) and a y-component (vertical ball speed due to ball bouncing machine). v_y^2 + v_x^2 = v_net^2....

So the horizontal speed affects the path length, but since it's orthogonal it doesn't change the time it takes the ball to complete it's vertical path. v_y (the vertical velocity of the ball) and d_y (the vertical distance the ball travels) don't change no matter how fast it's moving horizontally.

I'm not really sure what the whole point of the elaborate bouncing machine is here though... so I may well be misreading or misinterpreting something.