Formula: Impact Force
Impact Force is the slowdown of the racquet on impact (Impact Impulse), divided by the time in which it occurs (the dwell time). Impulse is the change in momentum, M(v1 − v2). From the formula for Work, we know v1, the velocity of the mass center (balance point) just before impact. Getting v2 requires application of the coefficient of restitution formula, converting mass center velocities (\(v\)) to impact point velocities (\(p\)) by multiplying by the ratio of d/r. Having (p1 - p2), we convert back by multiplying by \(r \div d\), to get (v1 - v2), then reduce and multiply by the racquet mass \(M\) to get the Impact Impulse, in kg·m/s. It turns out that Impact Impulse correlates closely with distance from the tip to the center of percussion. Dividing now by the dwell time, we get the Impact Force, in kg·m/s². This is so because of Newton’s Second Law, which says that a mass (kg) × an acceleration (m/s²) is a force. The Impact Force multiplied by its lever arm gives Torque, exactly. Impact Force correlates exactly with Impulse Reaction, and the closer the sweet spot is to the tip, the less the Impact Force.
\[ Work = \frac{1}{2} Mv_1^2 \]
\[ Work ≥ v_1 = \frac{2}{M} \times \sqrt{Work} \]
Converting v1, the mass center velocity, to initial impact point velocity p1:
p1 = (d/r)[(2/M)(Work)]½
From the coefficient of restitution formula:
p2 - s2 = c(s1 - p1)
we can find p2
p2 = cs1 + s2 - cp1
Now knowing p1 and p2, we can find v1 - v2 by multiplying by the ratio of the mass center radius (r) to the impact point distance from the axis (d) and plugging in the derived values of p1 and p2:
v_1 - v_2 = (r/d) [p1 - p2]
= (r/d) [(d/r)[(2/M)(Work)]½ - cs1 - s2 + c((d/r)[(2/M)(Work)]½)]
Impact Impulse = change in racquet momentum = M (v_1 - v2)
Impact Impulse = M(r/d)[(d/r)[(2/M)(Work)]½ - cs1 - s2 + c((d/r)[(2/M)(Work)]½)]
Simplifying:
Impact Impulse = M{(1 + c)[(2/M)(Work)]½ - (r/d)(cs1 + s2)}
Impact impulse is measured in momentum units (kg·m/s) and if we divide impact impulse by the time it operates we will have kg·m/s², which by Newton’s Second Law must be a force, measured in Newtons of force (1 Newton = 0.225 lb). The time to divide by is the dwell time of the ball on the racquet, which we assume to be 0.004 second for all racquets being compared.
Impact Force = Impact Impulse ÷ dwell time
= (M / t)*{(1 + c)*[(2/M)*(Work)]½ - (r/d)*(cs1 + s2)}
Impact torque (Torque) is Impact Force × the lever arm on which this force is applied, which is r, the distance from the axis of rotation (the hand) to the racquet’s mass center, or balance point. It turns out that the impact torque found from this formula matches the Torque from the formula derived otherwise, so we have corroboration that the Work derivation is correct.
| \(A_x\) | = | 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 |