Momentum

The momentum of a body is the product of its mass and its velocity. It is a vector: it has the same magnitude, the same direction and the same sense as the velocity, but “weighted” by the mass. A slow lorry and a fast ball can have very different velocities, yet comparable momenta, because what matters is the product $m\vv{v}$: it is the measure of “how much motion” a body carries with it, and hence of how hard it is to stop.

Principle — Momentum

$\ev{\vv{p} = m\,\vv{v}}$

Momentum is the vector twin of kinetic energy. Both grow with mass and with speed, but in different ways: $\vv{p}$ is linear in $\vv{v}$ and carries with it the direction of motion, while $E_\text{cin} = \tfrac{1}{2}m|\vv{v}|^2$ is quadratic in the magnitude and has no direction. This difference is the deep reason why, in collisions, the two quantities give independent and complementary equations.

	Momentum $\vv{p}$	Energy $E$
Type	vector (several components)	scalar (a single number)
Definition	$m\vv{v}$	$\tfrac{1}{2}m\lvert\vv{v}\rvert^2$
Transfer	impulse $\vv{I} = \vv{F}\,\Delta t$	work $L = \vv{F}\cdot\Delta\vv{s}$

The best way to see the analogy is to think of each quantity as a “reservoir” that changes only as a result of external transfers: impulse fills or empties the reservoir of momentum, exactly as work (and heat) do with that of energy.

Momentum and energy are treated with the same logic: a “reservoir” that changes as a result of external transfers (impulse for $\vv{p}$, work/heat for $E$).

Key formulae

$\vv{p} = m\vv{v}$ $\vv{I} = \vv{F}\,\Delta t$ $\Delta\vv{p}_\text{sistema} = \sum\vv{I}_\text{esterne}$

Links

Topics: Momentum and collisions Concepts: Momentum · Kinetic energy

Related exercises: Elastotennis · Ranking kinetic energies · Mass that halves