The Ballistic Pendulum

A bullet fired into a hanging block makes it swing upward. The higher it swings, the faster the block (and bullet) must have been moving right after impact, and the faster that was, the faster the bullet must have been flying beforehand. You can't measure the bullet's speed directly, but you can measure how high the block rises, and that height is the thread that connects back to the original speed.

The problem splits into two separate physical events, each with its own conservation law. Stage one is the collision itself: the bullet embeds in the block in a perfectly inelastic collision, so momentum is conserved but kinetic energy is not — some of it converts to heat and deformation as the bullet tears into the wood. Use momentum conservation here to relate the bullet's incoming speed to the combined block-and-bullet speed just after impact. Stage two is the swing: once the block and bullet move together, no more energy is lost to deformation, so mechanical energy is conserved as kinetic energy converts to gravitational potential energy at the top of the swing. Use energy conservation here to relate that same combined speed to the height risen.

This topic is where momentum and energy conservation meet inside one scenario, which is exactly the point of the unit: knowing which law applies to which moment. It also sets up the extension case where the bullet passes through the block instead of stopping inside it, still a momentum problem at the collision but now with two separate exit speeds instead of one combined one.