xfirebladex is assuming the acceleration from the throw, the acceleration of the diabolo from the start of the throw through to when you release it

Then we get into "jerk" (or "jolt" for some of you UKers) and so on... timing the throw is much simpler. Regarding a point made earlier about not knowing when it will reach its peak:The diabolo will reach its peak at exactly half the total time and be caught with the same speed as it was thrown with (assuming no air friction).

h = v t - 1/2 g t^2

h = t ( v - 1/2 g t )

v = 1/2 g T

H = 1/2 g T ( 1/2 T ) - 1/2 g ( 1/2 T )^2H = g / 8 * T^2