Thinking Inside the Box: The Mathematics of a Tennis Serve Roland Minton and Jake Bennett Roanoke College minton@roanoke.edu

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

Magnus Force Topspin Spin Vector Magnus Force

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

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

Basic Equations

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

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

Service Height Increase with higher ball toss and jump

Service Speed Increased speed yields less accuracy (shown: 90-134 mph)

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

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

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

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.