The Mathematics of Tennis

Projectile motion

This investigation requires the knowledge of parametric equations, vectors, basic geometry and the fundamental understanding of the game of tennis. In undertaking this task successfully students will gain an understanding of projectile motion as an application of a study of non-uniformly varying quantities.

Part A While practising their tennis Michael serves the ball from the baseline down the centre line to Tony, such that the path is represented by the equations for x(t) and y (t) , the horizontal and vertical distances in metres, t seconds after the serve. Also x(t) = 30t and y(t) = 3 – 3t – 4.9t2 t ≥ 0 sec

The important features of the court are that it is 23.774 m long and the net is 0.9m high and the service line is 6.401 m from the net.

Questions to Part A a) Find the initial position of a ball and explain the significance to the practical context of the tennis game.

b) Given that the velocity vector of the ball is represented by v(t) = [ x’(t), y’(t) ] and the speed of the ball at time t is the modulus of v(t). Find the speed at which Michael hits the ball.

c) Assuming the ball clears the net determine whether the ball will land inside the service area

d) Find the time t that the ball would be passing over the net and hence calculate if the serve will clear the net or not.

e) Discuss some assumptions and limitations of this model in relation to factors that exist in real life.

Part B

Using general equations for the horizontal and vertical position in flight x(t) = At + B and y(t) = at2 + bt +c where A, B , a, b, c are constants.

a) Find expressions for

i) The initial position of the ball ii) The initial speed of the ball

b) Use a graphics package to investigate the changes to the curve when parameters change. Discuss your findings and support your discussion with appropriate graphs.

c) The coach is trying