Formula: Work
Work is the joules of energy expended in accomplishing the stroke specified by the initial and final ball speeds, and the other constraints in the benchmark conditions.
Work is equal to the kinetic energy of the racquet mass center just before impact, which is ½ Mv1², given a certain ball speed before and after. This kinetic energy of the racquet is the result of the effort, or Work, put in by the player. The less Work needed to produce a certain ball speed, the better. A Work formula requires application of Conservation of Angular Momentum, as with Torque. Also, as in the Shock derivation, the coefficient of restitution gives the key to finding the linear velocity of the mass center (v1) before impact, and once that is known, a formula for Work follows by the formula for kinetic energy.
Power is the rate of doing work, joules per second, and one horsepower is 746 joules per second (watts), or 550 foot pounds per second.
Applying Conservation of Angular Momentum:
\[ Iw_1 + dbs_1 = Iw_2 + dbs_2 \]
Note that this involves the same assumption as in the Torque derivation about the restoring torque of the player being a non-impulsive force that may be disregarded in the conservation equation. Since \(d\) and \(I\) are in centimeters (see the Torque derivation):
\[ p_1 - p_2 = \frac{d^2b}{I}(s_2 - s_1) \]
\[ p_2 = p_1 - \frac{d^2b}{I}(s_2 - s_1) \]
\[ s_2 - p_2 = c(p_1 - s_1) \]
\[ (1 + c)p_1 = s_2 + cs_1 + \frac{d^2b}{I}(s_2 - s_1) \]
\[ p_1 = \frac{1}{1 + c} \times \left[s_2 + cs_1 + \frac{d^2b}{I}(s_2 - s_1)\right] \]
\[ v_1 = \frac{r}{d} \times p_1 = \frac{r}{d(1 + c)} \times \left[s_2 +cs_1 + \frac{d^2b}{I}(s_2 - s_1)\right] \]
\[ Work = \frac{M}{2}v_1^2 \]
\[ Work\ (joules) = \frac{Mr^2}{2}\left(\frac{1}{d(1 + c)} \times \left[s_2 + cs_1 + \frac{d^2b}{I}(s_2 - s_1)\right]\right)^2 \]
| \(A_x\) | = | 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 |