Principle

Conservation of mechanical energy

(M3.3.1) M3.3 · Conservation of Energy →

K_{i} + U_{i} = K_{f} + U_{f}

Statement

Kinetic plus potential energy at one instant equals kinetic plus potential energy at any later instant, provided no non-conservative force (friction, drag, an applied push) does work.

Symbols and units

$K_{i}, K_{f}$	initial, final kinetic energy	J
$U_{i}, U_{f}$	initial, final potential energy	J

When it applies

No non-conservative force (friction, drag, an external push) does work on the system.

Derivation

Combine the work-energy theorem with the definition of potential energy as the work done against a conservative force:

Δ K = W_{net} = - Δ U

Rearranged, this states that the sum K + U does not change:

K_{i} + U_{i} = K_{f} + U_{f}

Worked examples

Speed at the bottom of a frictionless ramp

A block starts from rest at height h = 3.0 m:

m g h = \frac{1}{2} m v_{f}^{2} \Rightarrow v_{f} = 2 g h

v_{f} = 2 (9.8) (3.0) \approx 7.7 m/s

Mass cancels — the speed at the bottom does not depend on how heavy the block is.

Watch out for

With friction or another non-conservative force present, mechanical energy is not conserved — the work it does (usually negative) must be added explicitly.

Connects to

Work-energy theorem