Coefficient of restitution

Also called Restitution coefficient

The coefficient of restitution, written e, is the ratio of the relative speed at which two objects separate after a collision to the relative speed at which they approached. It runs from 1 for an elastic collision down to 0 when the objects stick together.

AP Physics does not use this term. The word restitution appears nowhere in the 2024 Course and Exam Descriptions for AP Physics 1, AP Physics 2, AP Physics C: Mechanics or AP Physics C: Electricity and Magnetism, and it is on none of the four equation sheets. AP classifies collisions with three named cases instead, by what happens to the system's kinetic energy: elastic, inelastic and perfectly inelastic. If an exam question asks you to classify a collision, that is the vocabulary it is looking for.

It is still a real quantity, standard in engineering and in many introductory textbooks. It compares how fast the objects leave to how fast they arrived:

e=v2f−v1fv1i−v2ie = \frac{v_{2f} - v_{1f}}{v_{1i} - v_{2i}}
  • e=1e = 1: kinetic energy is conserved, the elastic case.
  • 0<e<10 < e < 1: some kinetic energy is lost, the inelastic case.
  • e=0e = 0: the objects separate at zero relative speed, so they travel together, the perfectly inelastic case.

For a ball dropped on a rigid floor the ratio of speeds becomes a ratio of heights, e=hbounce/hdrope = \sqrt{h_{\text{bounce}}/h_{\text{drop}}}, which is how it is usually measured in a lab.

Knowing ee costs nothing and makes the elastic-to-inelastic scale concrete. Just write your AP answers in the three-case language above.

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