Conservation of Energy

The principle of conservation of energy — energy cannot be created or destroyed, only transferred from one form to another. The total energy of a closed system is constant.

Dissipation — energy transferred to forms that are no longer useful, usually heating the surroundings through friction, drag or sound. The energy has not been lost or destroyed, but it has been spread out so thinly that it cannot be recovered. "Wasted" is a statement about usefulness, not about accounting.

Example: a ball dropped onto the floor bounces back lower than it started. Falling, its potential energy becomes kinetic. During the bounce the ball deforms, and some energy is transferred to internal energy and sound. It leaves the floor with less kinetic energy, so it can only rise to a lower height. Every joule is still accounted for — just not in a form that puts the ball back where it began.

Using conservation to solve problems — equate the energy at the start with the energy at the end, including anything dissipated. This is often much quicker than forces and suvat, because it skips the details in between.

Example: a pendulum released from 0.20 m above its lowest point. Rather than analysing the varying tension and the changing direction of motion, write mgΔh = ½mv², cancel the mass and get v = √(2 × 9.81 × 0.20) = 2.0 m s⁻¹ in one line.

When energy methods are the wrong tool — they give speeds but not times or directions, because energy is a scalar and carries no direction. If a question asks how long something takes or which way it ends up moving, you need forces and suvat, or momentum.

Efficiency in a real system —

efficiency = useful output ÷ total input

A Sankey diagram shows this: the width of each arrow is proportional to the energy it represents, so the total width of the outgoing arrows always equals the incoming one. The picture makes conservation visible — the arrows branch, but no width appears or disappears.