Formulas for evaluating racquet performance

The math involved in the derivations is high school algebra. The conservation principles applied are first year physics.

• Conservation of angular momentum
• Conservation of linear momentum
• Axis of rotation
• Benchmark conditions for objective racquet comparisons
• Coefficient of restitution
• Elbow Crunch (Elbow Yank)
• Impact Force (impulse force)
• Impulse Reaction
• Kinetic energy
• Moment
• New Balance Point
• The Parallel Axis Theorem (P.A.T.)
• Shock
  • Note regarding v = (r ÷ d)p
• Shoulder Crunch
• Shoulder Pull
• Sweet Spot
• Tip Speed
• Torque
• Torsion (Longitudinal torque)
• Units of measurement and conversion factors
• Variables
• Work (kinetic energy)
• Wrist Crunch

---

$$\begin{array}{}\text{(1)}& ShoulderCrunch(Newtons)=\frac{2}{0.01r+0.61}\times Shock\end{array}$$
Derivation

$$\begin{array}{}\text{(2)}& ElbowCrunch(Newtons)=\left(\frac{2}{0.01r+0.36}\right)\times Shock\end{array}$$
Derivation

$$\begin{array}{}\text{(3)}& WristCrunch(Newtons)=\left(\frac{2}{0.01r+0.08}\right)\times Shock\end{array}$$

$$\begin{array}{}\text{(4)}& TipSpeed(m/s)=\sqrt{\frac{2}{M}\times Work\times {\frac{e}{r}}^2}\end{array}$$
$e$ = distance from axis to tip, in cm Derivation

The $ImpactForce$ in Newtons acting at mass center due to impact.

$$\begin{array}{}\text{(5)}& ImpactForce=\frac{M}{t}\times [(1+c)\times \sqrt{\frac{2}{M}\times Work}-\frac{r}{d}\times (c{s}_1+s_2)]\end{array}$$
Derivation

Legend of Variables

$A_x$	=	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
$\omega$	=	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

See also: Units of measurement and conversion factors.