Formula: Coefficient of restitution
The coefficient of restitution (denoted by the symbol \(c\) in our formulas) is a measure of the elasticity of the collision between ball and racquet. Elasticity is a measure of how much bounce there is, or in other words, how much of the kinetic energy of the colliding objects before the collision remains as kinetic energy of the objects after the collision. With an inelastic collision, some kinetic energy is transformed into deformation of the material, heat, sound, and other forms of energy, and is therefore unavailable for use in moving.
A perfectly elastic collision has a coefficient of restitution of 1. Example: two diamonds bouncing off each other. A perfectly plastic, or inelastic, collision has \(c\) = 0. Example: two lumps of clay that don’t bounce at all, but stick together. So the coefficient of restitution will always be between zero and one.
The coefficient of restitution is the ratio of the differences in velocities before and after the collision. In other words, the difference in the velocities of the two colliding objects after the collision, divided by the difference in their velocities before the collision. In symbolic language:
\[ c = \frac{s_2 - v_2}{v_1 - s_1} \]
| \(c\) | = | coefficient of restitution |
| \(v_1\) | = | linear velocity of the racquet mass center before impact |
| \(s_1\) | = | linear velocity of the ball before impact (will be negative according to our convention that away from the player is positive) |
| \(v_2\) | = | linear velocity of the racquet mass center after impact |
| \(s_2\) | = | linear velocity of the ball after impact |
From the coefficient of restitution formula, it follows that
\[ s_2 - v_2 = cv_1 - s_1 \]
To find the coefficient of restitution in the case of a falling object bouncing off the floor, or off a racquet on the floor, use the following formula:
\[ c = \sqrt{\frac{h}{H}} \]
| \(c\) | = | coefficient of restitution (dimensionless) |
| \(h\) | = | bounce height |
| \(H\) | = | drop height |
For the Benchmark Conditions, the coefficient of restitution used is 0.85 for all racquets, eliminating the variables of string tension and frame stiffness that could add or subtract from the coefficient of restitution.
| \(A_x\) | = | Impulse Reaction, the translational force acting at the axis of rotation due to impact, in Newtons. Note that when \(d\) = \(q\) (\(q\) is the distance from the axis of rotation to the center of percussion), the expression within the second set of parentheses becomes zero. |
| \(a\) | = | linear acceleration of the mass center, in m/s² |
| \(b\) | = | mass of the ball, in kg |
| \(c\) | = | coefficient of restitution of the racquet/ball system |
| \(d\) | = | distance from the axis of rotation to the impact point, in cm |
| \(e\) | = | the distance from the axis of rotation to the tip |
| \(F\) | = | force applied at mass center, in Newtons |
| \(I\) | = | moment of inertia (swing weight) of racquet, in kgf/cm² |
| \(I_5\) | = | moment of inertia (swing weight) of racquet at 5cm from the butt, in kgf/cm² |
| \(I_7\) | = | moment of inertia (swing weight) of racquet at 7cm from the butt, in kgf/cm² |
| \(I_{10}\) | = | moment of inertia (swing weight) of racquet at 10cm from the butt, in kgf/cm² |
| \(I_a\) | = | moment of inertia (swing weight) of racquet at distance \(a\) from the butt, in kgf/cm² |
| \(M\) | = | mass of the racquet, in kg |
| \(m\) | = | mass in kg |
| \(ω\) | = | angular velocity of racquet, in radians/s |
| \(p\) | = | linear velocity of impact point, in m/s |
| \(r\) | = | distance in cm from mass center (balance point) to axis used in the stroke |
| \(s\) | = | ball velocity, in m/s (positive is away from player) |
| \(s_1\) | = | velocity of ball before impact, in m/s |
| \(s_2\) | = | velocity of ball after impact, in m/s |
| \(T\) | = | torque at axis of rotation, in Nms |
| \(t\) | = | dwell time, or duration of impact, in seconds |
| \(v\) | = | linear velocity of the mass center, in m/s |
| \(v_1\) | = | linear velocity, just before impact, of racquet mass center, in meters/second |
| \(v_2\) | = | linear velocity, just after impact, of racquet mass center, in meters/second |