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.
SportSim Sample
You Do Not Have To Be 9 Feet Tall To Get A No Spin Serve In
|
Another way that you can demonstrate a ball is
influenced by gravity is with the following small experiment.
Here's what you need:
a board (at least 6 feet long)
paper (enough to cover the board)
transparent tape
a marble
washable finger-paint or ink
stop watch or watch with a second hand
2 thick books or 2 bricks
pencil or pen
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
