BHS -> Mr. Stanbrough -> Physics -> Mechanics -> Momentum -> this page
Since momentum depends on mass and velocity (momentum = (mass)(velocity)), to say that you want maximum possible velocity for the ball is the same as saying that you want the largest possible momentum for the ball as it leaves the racket. Now the ball has essentially zero momentum when you hit it, since it is moving very slowly just before the racket hits it. You want to change this momentum to a very large momentum toward the other side of the net. So, to say that you want the largest possible momentum for the ball as it leaves the bat means that you want the largest possible change in momentum for the ball.
Aha! The impulse-momentum equation talks about change in momentum! It says that the change in momentum of the ball equals the impulse that you apply (with the racket) to the ball. So, to get the largest possible change in momentum, we want to apply the largest possible impulse to the ball.
Impulse depends directly on the force applied and the time the force is applied. (Impulse = (force)(time)). So, to get the largest possible impulse you should either:
So, swinging harder will hit the ball harder. (Duh?) Certainly, you want to apply maximum force by hitting the ball hard. If you hit the ball with twice the force, you will impart twice the impulse to the ball. Since impulse = change in momentum, this will double the ball's change in momentum. Since momentum equals mass times velocity, doubling the ball's momentum will double its velocity. However, if you try to apply too much force your coordination and timing will suffer, and your serve will not be accurate - you may even miss the ball!
You can also increase the impulse on the ball by increasing the time that the racket exerts its force on the ball - "following through". If you hit the ball for twice as much time, you will impart twice the impulse to the ball, which means twice the change in momentum for the ball. So, following through is important.
Of course, if you hit the ball hard and follow through, you will impart the greatest impulse to the ball. If you double both the force and the time, you get four times the impulse, and four times the change in momentum!
The same analysis would apply to hitting a golf ball, baseball, softball - whatever: