Momentum and conservation of linear momentumEdexcel A-Level Physics: Revision notes

The momentum of an object is the product of its mass and velocity: p = mv. It is a vector, in the direction of the velocity, with units kg m s⁻¹ (or N s). A 0.60 kg trolley moving at 1.5 m s⁻¹ has momentum 0.90 kg m s⁻¹.

Because momentum is a vector you must use a sign convention. If right is positive, an object moving left has negative momentum. Two objects with equal and opposite momenta have total momentum zero.

The principle of conservation of linear momentum states that, for a closed (isolated) system, the total momentum in a given direction before an interaction equals the total momentum in that direction after it. A closed system is one with no external resultant force acting on it.

In collisions and explosions the objects exert forces on each other (internal forces), which change the individual momenta but not the total. Friction, air resistance and other external forces must be negligible, or the time of the interaction very short, for the principle to apply in practice.

Newton's second law states that the resultant force equals the rate of change of momentum, F = Δp/Δt. When two objects interact, Newton's third law says they exert equal and opposite forces on each other, for exactly the same time. So their changes of momentum, FΔt, are equal and opposite, and the total momentum does not change.

This is why conservation of momentum follows from the third law and is useful: it lets you analyse an interaction without knowing the details of the forces.

For two objects with masses m₁ and m₂: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂, with velocities signed.

Worked example. A truck of mass 8000 kg at 3.0 m s⁻¹ hits a stationary 12 000 kg truck and they couple. 8000 × 3.0 = (8000 + 12 000)v, so v = 1.2 m s⁻¹. The first truck's momentum changes by 8000 × (1.2 − 3.0) = −14 400 kg m s⁻¹ and the second's by +14 400 kg m s⁻¹.

If objects stick together, treat them as one mass after the collision.

If objects are initially at rest and then fly apart (an explosion, a gun recoil, a person throwing a ball on a skateboard), the total initial momentum is zero, so the final momenta are equal and opposite.

Worked example. A 58 kg student and skateboard throw a 4.0 kg ball at 5.0 m s⁻¹: 0 = 4.0 × 5.0 − 58v, so the student recoils at v = 0.34 m s⁻¹. The smaller mass moves faster. The recoil is also an application of Newton's third law: the ball pushes back on the thrower.