1. Thinking Inside the Box: The Mathematics of a Tennis Serve
Roland Minton and Jake Bennett
Roanoke College

2. The Basics Forces on the tennis ball Gravity Air drag Magnus Force
Velocity
Air Drag

3. Magnus Force
Topspin
Spin Vector
Magnus Force

4. The Court Land inside service box and clear net. 3.5 ft 3 ft 17 ft

5. The Court
Land inside service box and clear net.
Angular acceptance

6. Basic Equations

7. A Flat Serve at 134 mph
0,0,3 v=55 w=100 a=-7 b=-12

8. Topspin For flat serve, angles are 7.3 and 8.7 deg.
Increasing spin leads to an increase in the angular acceptance window

9. Service Height
Increase with higher ball toss and jump

10. Service Speed
Increased speed yields less accuracy (shown: mph)

11. Initial Position
Hitting the ball inside the baseline increases the angular acceptance window about 0.4 degrees per foot
Increasing horizontal position increases the window about 0.3 degrees with maximum topspin from doubles sideline

12. How to Get Topspin (without really trying)
Toss the ball high; downward velocity imparts topspin as ball rolls on racket.
From The Physics and Technology of Tennis by Brody, Cross and Lindsey

13. q So H=1 m gives 53 rad/s, 15% increase in window Vel ball
Velocity of racket
So H=1 m gives 53 rad/s,
15% increase in window

14. Hit Up or Down? Flat serve: 7.3 – 8.7 down
During impact, racket rotates about 15 deg.
Downward velocity of ball lowers trajectory
Ball comes off racket about 12 deg lower than angle of racket at initial impact.
So the answer is Yes.