Dropping a ball in a train moving close to the speed of light? Suppose a train is moving very close to the speed of light, say 0.999c relative to a stationary observer on Earth.
Now a stationary observer on Earth will observe clocks on the train to tick slower than usual. 
Now suppose a boy within the train drops a ball.
The stationary observer measures how long it takes for the ball to hit the ground.  Will the result simply be $\sqrt{\frac{2s}{g}}$ or will the result be a greater number due to time dilation?
 A: Rely on the invariance of the interval $\tau$ between the ball being dropped and the ball hitting the floor. Also using $c=1$ units to keep the typing down.
The moving frame is un-primed and the stationary frame primed.
$$ t^2 - h^2 = \tau^2 = {t'}^2 - [{h'}^2 + (vt')^2] \,.$$
Because the velocity $v$ is horizontal both parties measure the same $h$, allowing us to write
$$\begin{align*}
{t'}^2 - [h^2 + (vt')^2] - \tau^2 &= 0 \\
{t'}^2(1 - v^2) - h^2 - \tau^2 &= 0 \\
{t'}^2(1 - v^2) - h^2 - (t^2 - h^2) &= 0 \\
{t'}^2(1 - v^2) - t^2 &= 0 \\
t' = \frac{t}{\sqrt{1 - v^2}} \,,
\end{align*}$$
so I owe Mew an apology for my comments in the earlier thread, but it does not violate the equivalence principle because we didn't use any information about the value of $h$ in the above work, so it works fine when $h=0$ as in the elevator case.
A: 
The stationary observer measures how long it takes for the ball to hit the ground. Will the result simply be $\sqrt{\frac{2s}{g}}$ or will the result be a greater number due to time dilation?

Of course, the the result will be time dilated and greater than $\sqrt{\frac{2s}{g}}$. It must be dilated if SR is a valid theory. You can simply consider the movement of the ball a kind of clock.
EDIT: The situation is equivalent to a similar problem. The boy is holding the ball in his hand. He lets it drop, bounce off the ground and come back to his hand (presuming it would return to the same height). Let's assume we cannot see the very ball (so we do not worry about the vertical movement), but we can see the boy open his hand (letting the ball drop), and then close his hand (to catch the ball when it is back).
Will the movements of his hand be time-dilated? Of course they will - according to SR - like everything else he does.
