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 times linear velocity of the racquet mass center before collision (Mv1)


ball mass times linear velocity of the ball before collision (bs1)


Impulse Reaction (Ax) times time of its operation (t)


racquet mass (M) times linear velocity of racquet mass center after collision (Mv2)


ball mass (b) times 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

Go to Derivation of Impulse Reaction formula.