Note regarding v = (r/d)p

As Voltaire remarked, the best is the enemy of the good. In other words, cavilling sticklers can obstruct significant improvements by insisting on perfection in trivial things.

Such, I fear, will be the case with the derivations offered in this site. Let a candid world judge the merits.

There is an error factor in the assumption that v1 = (r/d) p1, arising from the fact that the axis of rotation is moving, so there is a translational as well as a rotational component to the mass center velocity. However, the magnitude of this error factor is negligibly small.

The error factor is [a - (r/d)a] (a is the translational velocity of the hand (axis of rotation), r and d are the distances from the axis of rotation to the mass center and impact point, respectively).

The velocity of the impact point (p) is the sum of two components: one is the translational velocity of the axis of rotation (let’s call it a), and the other is the velocity due to rotation, p - a, which is dw (d is the distance from the axis of rotation to the impact point, w is the angular velocity). Likewise, v is the sum of a and a rotational component. The translational component of v is the same as that for p, but the rotational component is smaller because r, the distance from the axis of rotation to the mass center, is smaller than d.

v = a + (r/d)(p - a)

v = [a - (r/d)a] + (r/d)p

Despite the existence of this error factor, [a - (r/d)a], we can disregard it in the derivations on the following grounds:

For the First Benchmark Condition, the 80 mph return of a 110 mph serve, the translational component of the stroke is small because there is no time to wind up for a big stroke. The stroke is the slow sweep of an extended arm pivoting at the shoulder or even farther back, at the spine, so the hand speed in meters/s will be the sweep speed (in radians per second) times the distance (in meters) of the hand to the pivot point. This will be a small a, about 1 meter/second or 2 mph.

For the Second Benchmark Condition, the 135 mph serve, just before impact, a, the velocity of the hand, becomes small relative to the rotational component. This is because the racquet is whipping around the axis of rotation (hand), and v1 is increasing rapidly due to angular acceleration about the axis of rotation in the hand. During the stroke leading up to the wrist snap, the hand does acquire considerable velocity, but once the arm is fully extended the hand slows down, joining the sweep of the arm (see above for the small speed of this) and becomes a pivot point for the racquet as it whips rapidly around.

The error factor [a - (r/d)a] will be even smaller than the already small a, especially where the impact point is close to the mass center (as in a golf club). In the case of a tennis racquet, where (r/d) typically is about ½, the error factor is about half a.

During the impact, a becomes very small as the Impulse Reaction keeps the axis steady. So v2 will be almost exactly (r/d)p2.

Thus is the assumption that v = r/d p justified: the result is not perfection, but it is close enough for our purposes. Perhaps those who are dissatisfied with the completeness of the derivations offered here might offer some constructive supplementation.

To have included the error factor would have increased the complexity of the formulas beyond reason, and would have introduced a new variable a, hand speed, which depends on the build of the player. The derivations of formulas for Shock and Work concern only the racquet, and not the arm or the other parts of the body.