Elastic Potential Energy in a Spring

A spring holds energy even when nothing is moving because energy isn't just about motion, it's about configuration. Pulling or pushing a spring away from its natural length forces its coils to resist, and that resistance means you had to do work to get it there. That work doesn't vanish. It gets stored in the spring's stretched or compressed shape, ready to push or pull something once you let go. This is elastic potential energy, given by PE_elastic = (1/2) k x², where k is the spring constant (how stiff the spring is) and x is the displacement from the spring's rest length.

The x² is the detail worth sitting with. As you stretch a spring farther, the force needed grows too, since force is kx. So doubling the stretch doesn't just double the distance over which you push, it also doubles the force you're pushing against. Work is roughly the product of force and distance, so doubling both multiplies the stored energy by four. That's why stretching a spring twice as far stores four times the energy, not twice as much.

This topic plugs directly into the energy bookkeeping of the unit. Once you know how much elastic potential energy a spring stores, you can set it equal to kinetic energy, (1/2)mv², to find a launch speed, exactly like converting gravitational PE to KE in a falling object. Springs just give you a second way to load energy into a system before conversion, and bar charts treat PE_elastic as just another column that must balance against KE and other energy forms.