Vertical Springs

When a mass is dropped from rest onto a vertical spring, mechanical energy is conserved, transforming among gravitational potential energy, elastic potential energy, and kinetic energy. Both gravity and the spring force do work on the mass as it falls and compresses the spring.

Consider a mass $m$ dropped from rest at height $h$ above the natural length of a vertical spring with spring constant $k$. Set the reference point $y = 0$ at the natural length of the spring.

Phase 1: Dropping to Maximum Compression

Initially, the mass has gravitational potential energy $U_{g,i} = mgh$, zero kinetic energy, and zero elastic potential energy:

(Note we are assuming a total energy of $100\text{ J}$ for illustrative purposes here).

At maximum compression $y_{\max}$ below the natural length, the mass momentarily stops, so $K_f = 0$. The gravitational potential energy is $U_{g,f} = -mgy_{\max}$ (negative because the mass is below the reference), and the elastic potential energy is $U_{s,f} = \frac{1}{2}ky_{\max}^2$:

Applying conservation of mechanical energy:

Rearranging into standard quadratic form: