Conservation of Momentum in Collisions

Here is what Atlas has to say: Momentum is the product of an object's mass and its velocity: p = mv. It carries a direction, which means it is a vector quantity. When two objects interact in an isolated system — meaning no net external force acts on the system during the collision — the total momentum before the event equals the total momentum after. This is the Law of Conservation of Momentum. Why does it work? Think about Newton's Third Law. When puck A pushes on puck B, puck B pushes back on puck A with an equal and opposite force. Those internal forces cancel each other out when you look at the system as a whole. Impulse is the product of a force and the time it acts (J = F·Δt), and it equals the change in momentum. Because the two pucks feel equal and opposite forces for the same duration, the impulse each puck gains is equal and opposite, so the change in momentum of one exactly cancels the change in momentum of the other. Net change to the system: zero. In equation form for two objects: m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f Let's walk through a quick example. A 2.0 kg puck moving at 4.0 m/s east strikes a 1.0 kg puck at rest. Total momentum before = (2.0)(4.0) + (1.0)(0) = 8.0 kg·m/s east. Whatever happens during the collision, the total momentum after must also equal 8.0 kg·m/s east — we can then use the equation above, plus information about the collision type, to find the individual final velocities. In an elastic collision, kinetic energy is also conserved — think steel ball bearings in a Newton's cradle, a near-elastic model where energy loss to sound and heat is small. In a perfectly inelastic collision, the objects stick together and share one final velocity — think two football players colliding and moving as one unit. Momentum is conserved in both cases; kinetic energy is only conserved in the elastic case. Note the difference between collision types: in a perfectly inelastic collision the objects must stick together. A general inelastic collision is simply one in which some kinetic energy is lost — the objects may still separate after impact. Real systems are never perfectly isolated — friction, air resistance, and gravity can act — but when the collision time is very short, those external impulses are negligible and conservation of momentum is an excellent model. Hint: if you are ever unsure which direction to call positive, pick a direction, stick with it, and let the algebra handle the sign.