Hmmmm you could probably use one Arduino with two microswitches like these to stop and start the clock.
But by measuring the time, you're not going to measure the speed as such... you'll have to infer the speed from the Equations of Motion in which you would need to assume a value of g. (Or measure g in another experiment, like measuring the period of a pendulum.)
(Bit unclear as to your objective... your subject says speed, but your opening para says how fast it fell, which could be just the elapsed time? Time's one thing, measuring the terminal velocity is another)