The momentum of a system

Definition of momentum

In classical mechanics, the momentum (or impulse) of a material point of mass m and velocity v is the vector:

p = m * v

It has the same direction and sense as velocity, and its unit is kg·m/s. For a system of several bodies, the total momentum is the vector sum of the momenta of each body:

p_total = ∑(m_i * v_i)

Connection to Newton’s second law

Newton’s second law is expressed in its most general form as:

F = dp/dt

When the mass is constant, this simplifies to F = m * a. However, this formulation remains valid even if the mass varies (e.g. a rocket expelling fuel), which shows that momentum is more fundamental than velocity alone.

Isolated system

A system is said to be isolated (or pseudo-isolated) when the sum of the external forces acting on it is zero (or negligible compared to the internal forces, as during a very brief collision). It is this condition that allows us to establish the conservation of total momentum.

Example

A 2 kg cart is travelling at 3 m/s to the right. Its momentum is p = 2 * 3 = 6 kg·m/s, directed to the right (positive).

Quantity	Value
mass m	2 kg
velocity v	3 m/s
momentum p	6 kg·m/s

Common pitfall: momentum is a vector, not a scalar. Two objects of equal mass moving at the same speed in opposite directions have opposite momenta: their sum may be zero.