Tennis Directed Investigation
While practising their tennis, Ben serves the ball from the base line B, down the
centre line to James, such that the path of the ball is represented by equations for x
and y, the horizontal and vertical distances in metres, t seconds after the serve.
x = 30t and y = 3 – 3t – 4.9t2
B Net Service Line
A tennis court is 23.8 metres long with the net in the middle at a height of 0.9 m.
The service line 6.4 m from the net.
a) Find the initial position of the ball and explain its significance.
b) Find the speed at which Ben hits the ball.
c) Assuming that the ball clears the net, determine whether the ball will land
inside the service area (ie. before the service line).
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.
a) Using general equations for the horizontal and vertical position of the ball in
flight, find expressions for
i) The initial position of the ball
ii) The initial speed of the ball
iii) An equation which determines the time after serving when the ball will land.
b) The coach is trying to improve Ben’s serve by increasing the initial speed of the
ball by up to 5 kph and the height of the ball toss [and hence the height he hits the
ball at] by up to 10 cm.
i) Vary the equations for x and y to reflect this improvement.
ii) Verify the changes are within the coach’s limits by calculating the
initial speed and height of the toss for the new equations.
iii) Determine if the ball would still clear the net and also whether it
would land within the service area.
iv) Ben stands 2 m along the base line from the centre line. Determine
the maximum and minimum length of his serve if it is to land in the
v) Discuss the likely implications of altering his position on the base
line for service.
Improve or modify the mathematical model.
You could improve the model by taking into consideration some of the assumptions
or simplifications that exist in the current model.
You could modify the model by considering playing on a Half-court Tennis court, or
playing a social rather competitive game of tennis.
Show sufficient evidence that your model is realistic.