When a racket hits a tennis ball what force is exerted?

Answer 1

It is the same force but it is used more efficiently because the strings of the racket are put under tension like springs at the moment of hitting the ball, which with the ball's natural elasticity allows more energy to be transferred to the ball, making it go faster.

Answer 2

Momentum is mass of an object multiplied by its velocity and total momentum is conserved (much like energy). So when the racket hits the ball the amount of

momentum lost by the racket (it was moving with higher velocity during the swing)

is partially (consider mainly the elasticity of the ball and the energy spent on deforming it and the generated heat,sound etc.) gained by the ball causing the ball to have a change in its velocity (in terms of both direction and speed, note that speed is not velocity.) If the ball gets a rotation along with the linear velocity the the linear velocity will be smaller than it otherwise would. If you need to calculate the amount of force excerted on the ball at the time of impact, you need to solve I=integral(P*dt) where the integral is taken from 0 to end of time of impact. And I is basically calculated from (impact) the change in momentum of the racket. Recall though not all of this impact force is gonna be used up by the balls new velocity.