Momentum & Impulse

Momentum is p = m·v, a vector: the sign carries the direction along the line of motion. Impulse is the change in it, J = Δp = m(v₂ − v₁), and it equals the force multiplied by the time it acts, J = F·Δt. That last identity is the useful one: for a given change in momentum, a longer contact time means a smaller force, which is the whole principle behind crumple zones, airbags, catching nets and bending your knees on landing. The force reported is the average over the contact; the peak in a real impact is higher.

In a one-dimensional collision with no outside force, momentum is conserved: m₁v₁ + m₂v₂ is the same before and after. That alone is not enough to find two unknown velocities, so a second equation is needed — the coefficient of restitution e = (v₂′ − v₁′) ÷ (v₁ − v₂), which is 1 for a perfectly elastic collision, 0 when the bodies stick together, and between for everything real. The page solves both equations together: v₁′ = (m₁v₁ + m₂v₂ + m₂e(v₂ − v₁)) ÷ (m₁ + m₂), and the mirror image for v₂′.

Kinetic energy is conserved only when e = 1. Otherwise the difference leaves as heat, sound and permanent deformation — 86 % of it in a typical head-on car collision, which is exactly what a car is designed to do. The restitution coefficient is a measured property of the two materials and speeds, not something derivable: about 0.9 for a superball, near 0 for putty. Rotation, friction and two-dimensional geometry are outside this page. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.