Coefficient of Restitution - Equation

Equation

Picture a one-dimensional collision. Velocity in an arbitrary direction is labeled positive and the opposite direction negative.

The coefficient of restitution is given by

for two colliding objects, where

is the final velocity of the first object after impact

is the final velocity of the second object after impact

is the initial velocity of the first object before impact

is the initial velocity of the second object before impact

Even though the equation does not reference mass, it is important to note that it still relates to momentum since the final velocities are dependent on mass. It is one dimensional unitless parameter defined only along line of impact.

For an object bouncing off a stationary object, such as a floor:

, where

is the scalar velocity of the object after impact

is the scalar velocity of the object before impact

Coefficient of restitution is defined as negative of ratio of relative velocity of separation and relative velocity of approach along line of impact. It is never negative. The coefficient can also be found with:

for an object bouncing off a stationary object, such as a floor, where

is the bounce height

is the drop height

This is because energy is conserved, and so:

When the ball hits the floor, its gravitational potential energy (=mgh) is at a minimum, as h, height, cannot be any lower (i.e. the ball can't go through the floor), so all its energy is Kinetic. When kinetic energy=0 (i.e. at the peak of its motion when it momentarily stops), all the energy is potential. Thus the ratio of heights (maximum potential energy) is also a square-root ratio one:

For two- and three-dimensional collisions of rigid bodies, the velocities used are the components perpendicular to the tangent line/plane at the point of contact i.e. along line of impact.