Momentum and Collisions

What is momentum?

Momentum is mass in motion: p = m v, where m is the mass in kilograms and v the velocity in metres per second. Its unit is kg·m/s. Momentum is a vector: it points the same way as the velocity. A slow lorry and a fast tennis ball can have the same momentum; a lorry at speed has a great deal.

Impulse

To change an object’s momentum you must push on it for some time. The impulse is the force multiplied by the time it acts: J = F Δt = Δp, measured in newton seconds (N·s, the same as kg·m/s). The same change in momentum can come from a large force acting briefly or a small force acting for longer.

Conservation of momentum

During a collision, A pushes on B and B pushes back on A with an equal and opposite force (Newton’s third law), for exactly the same time. So their impulses are equal and opposite, and whatever momentum B gains, A loses. For anisolated system, one with no outside forces, the total momentum stays the same:

m_Av_A + m_Bv_B = m_Av_A′ + m_Bv_B′

The primes mark velocities after the collision. A useful consequence: the centre of mass of the system moves at a constant velocity, p_total/M, straight through the collision.

Kinetic energy in collisions

Kinetic energy, KE = ½ m v², is not a vector and is not always conserved in collisions. Some of it can become heat, sound and permanent changes of shape. Energy as a whole is still conserved; it just changes form. How bouncy a collision is can be described by the coefficient of restitution e: the speed at which the objects separate divided by the speed at which they approached.

Elastic, inelastic and sticky

In a head-on elastic collision between equal masses, the objects swap velocities: the moving one stops and the target moves off. When a heavy object hits a light one, the heavy one barely slows and the light one shoots off at up to twice the incoming speed. In a perfectly inelastic collision the objects move off together at v′ = (m_Av_A + m_Bv_B)/(m_A + m_B).

Collisions in two dimensions

Momentum is a vector, so it is conserved separately in each direction: the x-components add up to the same total before and after, and so do the y-components. In the momentum diagram the individual momenta are added tip to tail, and the resultant is identical before and after. Smooth pucks push on each other only along the line joining their centres at the moment of contact. In a glancing elastic collision between equal masses, one of which starts at rest, the two move off at right angles to each other, as snooker players know well.

Real-world applications

• Vehicle safety: crumple zones, airbags and seatbelts lengthen the collision time so that the same change in momentum produces a smaller force on the people inside.
• Sport: snooker, bowling and football all rely on momentum transfer; following through keeps the impulse acting for longer.
• Rockets and recoil: exhaust gases and a gun’s bullet carry momentum one way, so the rocket or gun moves the other.
• Forensic science: investigators work backwards from skid marks and final positions to the speeds before a crash.
• Particle physics: conservation of momentum reveals invisible particles in collisions at accelerators.

Common misconceptions

• “Momentum is lost in a crash.” Kinetic energy may be lost, but the total momentum of an isolated system is not.
• “Momentum and kinetic energy are the same thing.” Momentum is a vector (mv); kinetic energy is a scalar (½mv²).
• “The bigger object pushes harder.” The forces on the two objects are always equal and opposite; the lighter one simply changes velocity more.
• “Objects that stick together have lost their momentum.” They share it, and keep moving unless their momenta were equal and opposite.

What the model assumes

The pucks slide without friction or air resistance and do not spin. The table has no walls, so the two pucks form an isolated system; the simulation stops when they leave it. Each collision is treated as instantaneous, with the impulse acting along the line of centres and a constant coefficient of restitution. In the sticky collision the two pucks move off together at the centre-of-mass velocity; real objects that stick after an off-centre hit would also start to rotate. Puck sizes grow with mass only to make the difference visible.

Key takeaways

• Momentum p = mv is a vector; impulse J = FΔt equals the change in momentum.
• In an isolated system, total momentum is conserved in every collision.
• Kinetic energy is conserved only in elastic collisions; inelastic collisions turn some of it into other forms.
• Objects that stick together move at v′ = (m_Av_A + m_Bv_B)/(m_A + m_B).
• The centre of mass of an isolated system moves at constant velocity, even through a collision.