Question

A light spring of length l and rigidity k is placed vertically on a table. A small ball of mass m falls on it.

Answer 1

Equation of motion for the ball at the moment when the spring is compressed by Δx,

ma = mg - kΔx

As long as the acceleration of the ball is positive, its velocity increases. At the moment, when the acceleration vanishes, the velocity of the ball attains the maximum value. The spring is then compressed by an amount Δl given by

mg - kΔl = 0

Δl = mg/k

So when ball attains the maximum velocity, its height from the table

h = l - mg/k