Formula: Conservation of linear momentum
Momentum is the product of inertia and velocity. Inertia means the tendency of something not to change, and velocity means how fast it moves. So momentum means the tendency of an object in motion not to slow down.
Momentum is of two kinds, angular and linear. Both kinds are conserved in any collision. Conservation means that none is lost.
Linear momentum is the tendency of an object moving in a certain direction to keep going at the same speed in the same direction. It is the product of the object’s inertia (its mass M) and its velocity (v), or Mv.
Conservation of Linear Momentum, as applied to the collision of a tennis racquet with a ball, works like this:
Because of the conservation principle, we can write a before and after equation, setting the sum of momenta of the racquet and ball before the collision equal to the sum of momenta after the collision. The racquet’s momentum is the product of its mass (M) and the linear velocity of its mass center (v). The ball’s momentum is the product of its mass (b) and its linear velocity (s). Note that the mass centers of racquet and ball are not on the same line, so this is what is called an eccentric impact.
In the case of an eccentric impact such as this, there is a steadying force exerted by the player to keep the axis of rotation of the racquet from moving during the impact. The steadying impulsive force (known as Impulse Reaction) multiplied by the time of its operation, gives an additional momentum to add into the equation for momentum conservation in the chosen direction. Note that the only direction that matters here is the direction from the player to the net. As for signs, our convention is that a velocity into the player is negative, and away from the player is positive.
The conservation equation in words looks like this:
racquet mass × linear velocity of the racquet mass center before collision (Mv1)
plus
ball mass × linear velocity of the ball before collision (bs1)
plus
Impulse Reaction (Ax) × time of its operation (t)
equals
racquet mass (M) × linear velocity of racquet mass center after collision (Mv2)
plus
ball mass (b) × linear velocity of ball after collision (bs2)
In symbolic shorthand, the equation looks like this:
Mv1 + bs1 + Axt = Mv2 + bs2
With algebra, the Impulse Reaction can be found:
Mv1 + bs1 + Axt = Mv2 + bs2
Axt = Mv2 - Mv1 + bs2 - bs1
Ax = {M(v2 - v1) + b(s2 - s1)} / t
See the Impulse Reaction formula.
| \(A_x\) | = | 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 |