The coefficient of restitution e quantifies how “bouncy” a collision is. Combined with conservation of momentum, it completely determines the velocities after a collision.
Basic Knowledge
Definition of the coefficient of restitution:e = (v2 − v1) / (v1 − v2)
= relative velocity after collision (speed of separation) / relative velocity before collision (speed of approach)
Range of e: 0 ≤ e ≤ 1
・e = 1: perfectly elastic collision (magnitude of relative velocity conserved)
・e = 0: perfectly inelastic collision (objects stick together, relative velocity zero)
・0 < e < 1: inelastic collision (many real collisions)
Collision with a wall (the wall does not move):e = v' / v (v: speed before hitting the wall, v': rebound speed)
Find v1 and v2 by solving the conservation-of-momentum equation and the coefficient-of-restitution equation simultaneously.
Coefficient of restitution: e = (v2 − v1) / (v1 − v2)
e = 1: elastic collision (magnitude of relative velocity unchanged)
e = 0: perfectly inelastic collision (objects stick together)
Collision with a wall: e = |v'| / |v|
Simultaneous equations: conservation of momentum + equation for e → determine v1 and v2
Further Study (Background and Significance)
Transforming the definition of the coefficient of restitution gives: v2 − v1 = −e(v2 − v1). This means that the relative velocity after the collision is e times the relative velocity before the collision and points in the opposite direction. When e = 1, the magnitude of relative velocity is maintained (the equivalent condition for an elastic collision); when e = 0, relative velocity is zero (the objects stick together).
The coefficient of restitution varies with material, shape, and speed. For example, e ≈ 0.7 for a tennis ball, e ≈ 0.9 for a superball, and e ≈ 0 for clay.
The equation for e provides another way to handle elastic collisions without using conservation of energy. Conservation of momentum (one equation) + coefficient of restitution (one equation) = two equations for two unknowns. This method is easier than using the energy-conservation equation (a quadratic equation).
For a ball dropped onto the floor: when it is dropped from height h₀ and rebounds to height h', e = √(h'/h₀) (from the relationship speed ∝ √h).
- e = relative speed of separation / relative speed of approach
- 0 ≤ e ≤ 1
- e = 1 → elastic collision
- e = 0 → perfectly inelastic collision (objects stick together)
- Conservation of momentum + equation for e → two simultaneous equations
- Rebound from the floor: e = √(h'/h₀)
- The denominator is the speed of approach, and the numerator is the speed of separation
Points to Note (Common Confusions)
① Do not reverse the numerator and denominator (if e > 1, it is wrong). ② Substitute velocity signs and directions accurately. v1 and v2 in the definition are velocity components in the positive direction. ③ Even when e = 1, it is “equivalent to conservation of energy,” but using the equation for e is easier. ④ For a collision with a wall, substitute the wall's velocity as 0.
Practice
- An object moving at 3.0 m/s collides with an identical stationary object with coefficient of restitution e = 0.5. Find the velocities of both objects after the collision.
- A ball dropped from a height of 1.25 m rebounds to 0.80 m. Find the coefficient of restitution.
- Explain the definition of the coefficient of restitution using “relative velocity.”