Could you help me solve a physics problem? An axis of length L and mass M is balanced on the side of the table at its center of gravity. A ball with a relative height h is dropped onto its most protruding end. What is the speed of the ball immediately after impact?
Sure! Given the brief question I will simplify the calculation by assuming that there is no energy conversion from sources like heat dissipation, friction or noise. With that we can assume that the ball and the axis perform an elastic collision since both can move freely with the added complication that the axis is balanced on the table.

First we need to calculate the balls velocity at the time of impact (v1):

g = earth gravity 9.81m/s^2
v1 = sqrt(2*g*h)

Since the axis is balanced at the table edge it acts like a class 1 lever potentially resulting in a mechanical advantage.
Due to the balance in the center the mechanical advantage of that lever is 1 since both sides are equally long.
With an advantage of one we can just ignore the levering action and calculate a fully elastic collision.

v2 being the velocity after impact
mb being the mass of the ball
ma being the mass of the axis

v2 = (mb - ma) / (mb + ma) * v1

combined with the first equation the result is:

v = (mb - ma) / (mb + ma) * sqrt(2 * g * h)