Momentum

Linear momentum is a vector

The SI unit is kg m s⁻¹. A 2 kg object moving at +3 m s⁻¹ has p = +6 kg m s⁻¹; the same object moving at −3 m s⁻¹ has p = −6 kg m s⁻¹. Signs encode direction along the chosen axis.

Newton's second law can be written more generally as ΣF⃗ = dp⃗/dt. For constant mass, this reduces to ΣF⃗ = ma⃗.

Impulse changes momentum

For a constant force, J = FΔt. Airbags, helmets and padded mats increase the stopping time, allowing a given momentum change to occur with a smaller average force.

Conservation of momentum

For a system, internal forces transfer momentum between its parts. If the net external impulse is negligible over the interval of interest, total momentum stays constant:

This condition is often an excellent approximation during a brief collision even when external forces such as weight exist, because their impulse during the short collision is small.

The system boundary is crucial. Two colliding carts together may form an almost isolated system even though each cart individually experiences a large force from the other. Those interaction forces are internal to the two-cart system and cancel in the total momentum balance.

Elastic and inelastic collisions

Momentum conservation does not tell you whether kinetic energy is conserved. In an elastic collision, both total momentum and total kinetic energy are conserved. In an inelastic collision, momentum is conserved for an isolated system but some kinetic energy becomes deformation, thermal energy or sound.

If objects stick together, the collision is perfectly inelastic. They share one final velocity.

Set the signs before calculating

Choose a positive direction and keep it throughout the problem. A negative velocity does not mean “slower”; it means motion opposite the chosen positive direction. In two dimensions, conserve x- and y-components separately.

For two-dimensional collisions, draw the momentum vectors before writing equations. One scalar conservation equation is not enough: use one equation for x and another for y. If the final directions are unknown, geometry and kinetic-energy information may supply the additional constraints.

Worked examples

1. Momentum of a moving car

A 1200 kg car travels east at 15 m s⁻¹. What is its momentum?

2. Impulse from a force

A constant 40 N force acts east on a ball for 0.15 s. What momentum change does it produce?

3. Perfectly inelastic collision

A 2.0 kg cart moving at +4.0 m s⁻¹ sticks to a 3.0 kg cart initially at rest. Find their common final velocity.