Elastic Potential Energy

A compressed spring holds energy because something had to push it there. As you squeeze the spring from its natural length, you do work against the force it pushes back with, and that work doesn't vanish. It gets stored in the stretched or compressed bonds of the material, ready to be given back the instant the spring is released. Nothing is moving, but the configuration itself holds the energy, the same way a raised book holds gravitational potential energy just by being up high.

The spring force is F = kx, growing as you compress it further, so you can't just multiply force times distance like you would for a constant push. You have to add up the work done at every tiny slice of compression, from 0 to x. That's the integral of kx dx from 0 to x, which gives (1/2)kx^2. The one-half is the fingerprint of any force that grows linearly with distance: you're averaging a force that starts at zero and ends at kx, so the total work is half of what you'd get if the full force acted the whole way. This is exactly why doubling the compression quadruples the stored energy, not doubles it. x² means compression twice as large stores four times the energy, because you're pushing against a steadily increasing resistance over a longer distance too.

This sits next to gravitational potential energy as the second major form of stored energy in Unit 3, and both work the same way: potential energy is the negative antiderivative of force with respect to position. Once the spring releases, conservation of energy lets you set (1/2)kx² equal to the kinetic energy it creates, collapsing a messy force problem into simple algebra.