In this chapter we consider three classic problems in momentum and energy conservation.
The Ballistic Pendulum
A bullet is fired into a block of wood that is hanging by a string from the ceiling. The mass of the bullet and the block are given, as is the incoming speed of the bullet. Find the height to which the block + bullet swings before stopping.
This problem involves both momentum and energy, but it requires great care that we don’t invoke energy conservation when non-conservative forces are acting, and that we don’t invoke momentum conservation when external forces are acting. Basically the trick is to dance around the problem only using the right tool for the job. The bullet colliding with the wood block involves a non-conservative force (the friction that slows the bullet within the block), so the mechanical energy of the bullet + block system is not conserved. But the force between the bullet and the block is internal to the bullet + block system, so momentum is conserved (we assume that the bullet digs-in and lodges so quickly that the block doesn't yet swing very far, so the external force from the string has no horizontal component during the collision). So we use momentum conservation to determine the speed of the block + bullet as they begin to swing. Then once it is swinging, the tension force from the string kills momentum conservation, but it does no work (it acts perpendicular to the direction of motion), so mechanical energy is conserved, and we use that to find the height.