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July 2003 Article Tennis Server
Top Pros (Women)
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Years ago there was a misconception, perhaps more appropriately termed "tennis lore," that you had to be 9 feet tall to get a flat (no spin) serve in. With 6 ft. 10in. Croatian, Ivo Karlovic knocking out No. 1 seed Lleyton Hewitt at Wimbledon after the first round, I thought we would demonstrate the mathematics of this particular trajectory scenario. First of all this isn't true. You don't have to be 9 feet tall nor does your combined height including your extended arm, racquet, and jump height need to be 9 feet or more. This is because both gravity and aerodynamic drag act on the ball during its flight. Gravity accelerates the ball downwards while drag (air resistance) creates a retarding force slowing the ball's forward motion. This causes a curved trajectory. Where did the "9 feet" come from? If you look strictly at the geometry of a tennis court and the minimum angle it would take to hit the ball from the baseline, clear the net and land in the service court, you find that the ball needs to be hit from a height of about 9 feet. Here's how this works. Here is a diagram of a tennis court and its dimensions including the net. The scenario: you serve just outside the baseline; the ball just clears the net down the middle and lands just inside the service court. Draw a straight line from the court impact point, over the net and extend that straight line to the baseline. Now we'll use a little trigonometry to determine the height the ball was struck at to achieve this. There are several ways to do this. Perhaps the easiest is to just draw your diagram to scale. The distance from the baseline to the opposing service line is 60 feet. At the center, the net is 3 feet. The diameter of the ball is about 2.6 inches in diameter so we need at add at least an inch or two from the middle of the ball to make sure it clears the net. If you scale this correctly and then measure the height from the ground at the baseline to the point the straight line " trajectory" crosses the baseline, you get just about 9 feet.
However as you can see, this scenario does not take into consideration gravity, aerodynamic drag or even the velocity of the ball or the angle it is hit. A while back we had two articles regarding tennis ball trajectories called Trajectories 101A and Trajectories 101B. In particular, Trajectories 101B presented equations of motion which account for gravity and aerodynamic drag. The column mentioned a web-based interactive program which uses those equations that you can have some fun with, called SportSim. (It takes about 30 seconds for the page to load.) You can play with the values of speed and height and determine that you do not have to be 9 feet tall (108 inches) to get a no spin serve in. Here's an example of the ball struck at about 85 miles per hour and 92 inches high.
Another way that you can demonstrate a ball is influenced by gravity is with the following small experiment. Here's what you need:
Using the tape cover the board with the paper. Mark a starting line at the top of the board. Prop one side of the board up with the books or bricks. Dip one side of the marble (or ball) in the finger paint. As the ball rolls down the ramp the ball will make a pattern on the paper. Place the marble on the starting line. Don't push the ball - just let it go. Record the time it takes to make 2 marks on the paper. Then repeat the experiment for 4 marks, 6 marks, 8 marks, etc. Try it with a different ball - do the results change?
Graph your results. Place time on one axis and number of marks on the other. You don't get a straight line do you? This is because gravity acting on falling body increases the speed of the fall. This believe it or not is an experiment two time Nobel Prize winner Madame Marie Curie would do with her students. Okay, she didn't really do it using a tennis ball. However, Madame Curie's son-in-law Frederic Joliot (married to daughter Irene) was a tennis fanatic. Irene and Fred also were awarded Nobel prizes. They were both very athletic and Fred had a tennis court built at their home. (Fred HAD to win.) Fred would later work with the French underground during World War II, be appointed by Charles de Gaulle to head France's first Atomic Energy Commission and later head the Curie Institute. What a family! Hope you have enjoyed Wimbledon! Until Next Month ... Jani
This column is copyrighted by Jani Macari Pallis, Ph.D., all rights
reserved.
Dr. Jani Macari Pallis is the founder and CEO of Cislunar Aerospace,
Inc., an engineering and research firm in San Francisco. In addition
to her engineering practice, she has led two collaborations between
NASA and Cislunar, creating educational materials on the aerodynamics
of sports for pre-college students and educators. As the head of
NASA's "Aerodynamics in Sports" project, she has led a team of
researchers investigating the aerodynamics, physics and biomechanics
of tennis. The group has conducted high speed video data capture at
the US Open and research of ball/court interaction, footwork, serve
speeds, trajectories and ball aerodynamics. Pallis received a BS and
MS from the Georgia Institute of Technology, an MS in mechanical
engineering from the University of California, Berkeley and a Ph.D.
in mechanical and aeronautical engineering from the University of
California, Davis. She is a member of the Executive Committee of The
International Sports Engineering Association.
Questions and comments about these columns can be directed to Jani by
using this form.
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