Definition and momentum vector

What is momentum?

The momentum (sometimes called impulse, denoted by p) of a material point of mass m moving at a velocity v is defined as:

p = m * v

It is a vector quantity: it has the same direction and sense as velocity. Its unit in the International System of Units is kg·m/s.

Link to Newton’s second law

Newton’s original formulation relates the resultant force F to the change in momentum rather than to acceleration alone:

F = dp/dt

If the mass is constant, we obtain F = ma, but this general form remains valid even if the mass varies (rocket, treadmill, ejection of matter), which is not the case for F = ma.

Momentum of a system

For a system of n material points, the total momentum is the vector sum of the individual momenta:

P = ∑ (mi * vi), i = 1 to n

Example

A 2 kg cart is travelling at 3 m/s to the right (+x). Its momentum is p = 2 * 3 = 6 kg·m/s in the +x direction.

Common pitfall

Do not confuse momentum (a vector quantity, dependent on the direction of motion) with kinetic energy (a scalar quantity, always positive, given by v²). Two identical carts travelling in opposite directions have opposite momenta but equal kinetic energies.