Formula: Impulse Reaction
The formula for Impulse Reaction uses Conservation of Linear Momentum. Here we are considering linear, not angular, momenta with respect to only a certain direction, and not about an axis of rotation. The system momenta before impact, plus any external linear impulse (player push or pull on the axis of rotation, multiplied by the time it operates), equals the system momenta after impact — all along the direction from the player to the net. The player force is the Impulse Reaction applied by the player at the axis of rotation to restrain the axis from being yanked in this eccentric impact, and this Impulse Reaction operates for the dwell time of the ball on the racquet. Unlike Torque, Impulse Reaction is an impulsive force, so it must be included in the conservation equation. From this conservation principle we can write an equation and then solve for Impulse Reaction. Note that when Impulse Reaction is zero, the impact is at the Sweet Spot (center of percussion). The rankings under Impulse Reaction (positive = good, negative = bad) turn out to be the same as the rankings for Impact Force. So Sweet Spot location directly affects the amount of Impact Force.
\[ Impulse\ Reaction\ (Newtons) = \left(\frac{b(s_2 - s_1)}{t}\right)\left(1 - \frac{Mrd}{I}\right) \]
The impact of a ball with a racquet is a case of eccentric impact,
so both rotation (turning about the axis of rotation, at the middle of the hand) and translation (movement of the axis of rotation backwards or forwards) result from the collision. In other words, the hand will be both pulled and twisted when the ball hits the racquet. These resultant forces from impact should be minimized.
Impulse Reaction and Torque are the two resultant forces from an eccentric impact, such as the collision of a tennis ball with a racquet face.
Applying Conservation of Linear Momentum, the sum of momenta before impact, plus the Impulse Reaction (\(A_x\)) at the axis of rotation × the dwell time (\(t\)), will equal the sum of momenta after impact. Linear momentum is mass × velocity (\(Mv\)). In abstract terms, the conservation equation looks like this:
\[ bs_1 + Mv_1 + A_xt = bs_2 + Mv_2 \]
\[ A_xt = b(s_2 - s_1) + M(v_2 - v_1) \]
\[ A_xt = b(s_2 - s_1) - M(v_1 - v_2) \]
\[ (v_1 - v_2) = \frac{rdb}{I}(s_2 - s_1) \](from the Torque formula)
\[ A_xt = b(s_2 - s_1) - \frac{Mrdb}{I}(s_2 - s_1) \]
\[ A_x = \left(\frac{b(s_2 - s_1)}{t}\right)\left(1 - \frac{Mrd}{I}\right) \]
| \(A_x\) | = | 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 |