## Derivation of Impact Force formula

Work = ½ Mv1²

=> v1 = [(2/M)(Work)]½

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:

v1 - v2 = (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 (v1 - 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 times 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.