Energy, work, and conservation of energy and momentum

What the exam asks

Expect a collision, a recoil, a slope, or a machine with an efficiency, and numbers chosen to work without a calculator. The commonest items are: find the common velocity after two bodies stick together, decide whether a described collision was elastic, compare kinetic energy before and after, and work out a power or an efficiency from a table. The trap that catches most candidates is hearing 'inelastic' and concluding that momentum was lost. It was not. Kinetic energy went missing, momentum did not, and the momentum equation is still your first line. A second trap is treating a body that stops as evidence against conservation of momentum: friction is an external force, so the system you chose was too small, and the momentum went into the Earth.

This topic covers work, energy, power and momentum, and the two conservation laws that come with them. The formulae are supplied when they are needed. What has to be carried in is the judgement of which law applies to the situation in front of you, because the two laws have different conditions and the exam is built on the difference.

Momentum is conserved whenever no net external force acts on the system, which in a collision is essentially always: the impact forces are internal and cancel in pairs, and they are so large that gravity and friction do nothing measurable in the milliseconds of contact. Kinetic energy is a separate question. It survives only an elastic collision. In every other collision some of it becomes heat, sound and permanent deformation.

That gives the topic its central move. In a collision, write momentum conservation first, because it holds regardless. Then ask separately whether kinetic energy is conserved, and let the stem tell you: bodies that stick together have lost the most kinetic energy they can lose, bodies that bounce apart perfectly have lost none. Energy questions away from collisions are usually the same idea in a different costume, where the total is fixed and the accounting is between the stores.

What to hold

• Work is the force multiplied by the displacement along that force, so a force perpendicular to the motion does no work at all.
• The net work done on a body equals its change in kinetic energy, which is often faster than any force-and-acceleration route.
• Power is work divided by time, so doing the same job faster is more power and exactly the same work.
• Efficiency is useful energy out divided by total energy in, and the shortfall is not destroyed. It leaves as heat, sound or something else you were not counting.
• Kinetic energy goes as the square of speed, while momentum goes linearly with it, so doubling a body's speed doubles its momentum and quadruples its kinetic energy.
• Momentum is conserved in any collision where external forces are negligible, and that includes every inelastic collision.
• Inelastic does not mean momentum is lost. It means kinetic energy is not conserved. Momentum survives the collision either way.
• The kinetic energy missing from an inelastic collision has become heat, sound and deformation, so total energy is conserved even though kinetic energy is not.
• A perfectly inelastic collision is one where the bodies move off together, and it is the case that loses the most kinetic energy the momentum will allow.
• In an elastic collision both momentum and kinetic energy are conserved, which is what gives you a second equation and lets you solve for two unknown final velocities.
• Impulse is force multiplied by contact time and it equals the change in momentum, so spreading the same momentum change over a longer time lowers the force.
• With only conservative forces acting, the speed at the bottom of a frictionless slope depends on the height dropped and not on the mass or on the path taken to get there.
• An explosion or a recoil is a collision run backwards: momentum starts at zero and must still total zero afterwards, while kinetic energy appears from a store that was chemical.