elastic potential energy
Stretch a rubber band, pull back a bowstring, or press down on a trampoline, and you can feel it pushing back and storing your effort. Elastic potential energy is the energy held in something that has been stretched, squeezed, or bent out of its natural shape — energy that springs the object back and can be handed to whatever it launches.
For an ideal spring that obeys Hooke's law (the restoring force grows in proportion to the stretch, F = -k x, where k is the spring's stiffness and x is how far it is stretched or compressed from its natural length), the stored energy is U = 1/2 k x^2. The x^2 means stretching twice as far stores four times the energy. Geometrically it is the triangular area under the force-versus-stretch line — which is exactly why a variable force like a spring's needs this squared formula rather than a simple force-times-distance.
You meet it in bows and slingshots, in the springs of a pogo stick or a car's suspension, and at the heart of every oscillation — a mass on a spring trades elastic potential energy for kinetic energy and back, forever if nothing drains it. One honest limit: U = 1/2 k x^2 assumes the material is truly elastic and stays within Hooke's law. Stretch it too far and it heats up, deforms permanently, or snaps, and the neat formula no longer holds.
A spring with stiffness k = 200 N/m is compressed x = 0.10 m. Its stored elastic potential energy is U = 1/2 k x^2 = 1/2 x 200 x (0.10)^2 = 1/2 x 200 x 0.01 = 1 J. Release it against a ball and, ignoring friction, that 1 J becomes the ball's kinetic energy.
Squeeze the spring twice as far and you store four times the energy — the x^2 at work.
The 1/2 in 1/2 k x^2 is not optional: because the spring's force builds up from zero as you stretch it, the energy is the average force (half the maximum) times the distance, not the full force times distance.