Dynamics

A Collision Keeps Momentum, and the Restitution Coefficient Sets the Bounce

Momentum m₁u₁+m₂u₂=m₁v₁+m₂v₂, restitution e=separation/approach, elastic vs inelastic, KE kept (1+e²)/2

Two objects collide. The total momentum is always conserved through the collision, and how much they bounce is set by one number, the coefficient of restitution e, from e=1 (perfectly elastic) to e=0 (they stick).

Ball 1 approaches ball 2 at rest. The relative speed at which they close in is the approach speed. Drag v: for an elastic collision of equal masses, the two simply swap velocities.

Even with unequal masses, the total momentum before and after is the same. Drag the mass ratio: the after-velocities change, but the total of the momentum bars stays put (there is no outside force).

The coefficient of restitution e measures the bounce: e = separation speed ÷ approach speed. Drag e and the separation speed becomes e times the approach speed (e=1 bounces back unchanged, e=0 sticks).

Pick a collision type. Perfectly elastic (e=1) even conserves kinetic energy, inelastic (0<e<1) loses some, and perfectly inelastic (e=0) sticks and loses the most. Momentum is conserved in every case.

Drag e to see how much kinetic energy survives. In a head-on collision of equal masses the surviving fraction is (1+e²)2, so e=1 keeps all of it and e=0 keeps only half, the rest scattering into heat, sound, and deformation.

In PracticeTo sum up, in a collision the total momentum is always conserved as long as there is no outside force (m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂). How much they bounce is set by the coefficient of restitution e = separation relative speed ÷ approach relative speed. e=1 is perfectly elastic (kinetic energy conserved), e=0 is perfectly inelastic (they stick, maximum loss), and in between is partially elastic. The velocities after a collision come from solving the momentum equation and the restitution equation together. The key is that momentum is always conserved, but kinetic energy only when e=1. For a head-on collision of equal masses the surviving kinetic-energy fraction is (1+e²)2. From the next lesson we move from a point to a rotating rigid body, describing rotation with angular position, angular velocity, and angular acceleration.

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