conservation of mechanical energy
Drop a ball and it speeds up exactly as it loses height; throw it up and it slows exactly as it climbs. Behind this trade sits one of physics' most useful rules: conservation of mechanical energy. When only conservative forces (like gravity or an ideal spring) are doing work, the total of kinetic plus potential energy never changes — it just shifts back and forth between the two forms.
In symbols, K_i + U_i = K_f + U_f — the kinetic-plus-potential total at the start equals the total at the end. Equivalently, whatever kinetic energy is gained, an equal amount of potential energy is lost: ΔK + ΔU = 0. This holds precisely when no nonconservative force (no friction, no air drag, no push from outside the system) does any work along the way.
It is a spectacular shortcut. To find how fast a ball is going after falling a height h, you do not need to track time or acceleration — just set 1/2 m v^2 = m g h and solve, giving v = sqrt(2 g h). The honest caveat: real motion has friction and drag, which quietly siphon mechanical energy into heat, so the mechanical total does shrink. Then you must widen the ledger to the full conservation of energy, adding a dissipation term for the lost heat.
A pendulum bob is released from rest at a height 0.20 m above its lowest point. Ignoring air resistance, all its potential energy becomes kinetic at the bottom: m g h = 1/2 m v^2, so v = sqrt(2 g h) = sqrt(2 x 9.8 x 0.20) which is about 2.0 m/s — and notice the mass cancels out completely.
No mass, no stopwatch needed: energy conservation turns a drop height straight into a speed.
This shortcut is exact only if nonconservative forces do zero work. If friction or air drag is present, mechanical energy is not conserved — you must instead use the full conservation of energy and account for the heat generated.