Lab: Spring Launcher Energy Transfer

The setup is simple on purpose. A cart sits against a spring on a track; you compress the spring a measured distance x, let it go, and a photogate clocks the cart's speed v right as it leaves the spring. If the track is frictionless and the spring is ideal, all the stored spring potential energy becomes kinetic energy: one-half k x² equals one-half m v^2. That single equation is the whole lab. Measure x, measure v, and you can solve for k, or predict v for a new compression once you know k.

The clever part is how you handle the data. Instead of plotting v versus x, which gives a curve, you plot v² versus x^2. Rearranging the energy equation gives v² = (k/m) x², which is a straight line through the origin with slope k/m. That means a messy square-root relationship becomes something you can fit a ruler to: find the slope, multiply by the cart's mass, and you have k. A slope of 900, for example, with a 0.40 kg cart, gives k = 900 x 0.40 = 360 N/m.

This lab is where energy bar charts meet real measurement. If your measured launch speed is consistently lower than the one you predicted from k and x, energy conservation hasn't failed. Some spring potential energy is leaking into friction or heat instead of showing up as kinetic energy, so the kinetic energy bar is shorter than predicted and a thermal energy bar should appear in its place. That's the bridge into later topics on nonconservative forces and energy dissipation.