The Physics of Tennis
by Blake Sharin
Introduction
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The Sweet Spots
Vibration Node
Centre of Percussion
The Dead Spot
Coefficient of
Restitution
The Sweet Spot
• Two Sweet Spots
– Vibration Node
– COP
• Force Transmitted to
hand
– Motions of the handle
• Rotation
• Translation
• Vibration
• Maximum Speed of
Ball
Vibration Node
• Behaves like a Uniform
Beam
• Fundamental Mode
• Frequency
– 100Hz for flexible frame
– 140Hz for stiff frame
• Two nodes
– Near the center of strings
– Near the handle
Vibration Node (Contd.)
• Frequency is 2.75
times the fundamental
frequency
• Not excited with any
significant amplitude
– Duration, T=5ms
– Peaks at Zero at
f=1.5/T=300Hz
• Close to second mode
frequency
Centre of Percussion
• Known as the impact
point
• Conjugate points
– Impact near tip
• Axis of rotation is about
½ way between the end
of the handle and the
CM
– Impact near throat
• Axis of rotation is
beyond the end of the
handle
Power Swinging
• More power when you
swing at the ball near
throat of racquet.
• When ball hits near the
throat of racquet, the ball
will have more ball speed.
• Heavier the racquet,
more speed applied to
the ball
• Many types of racquets
Coefficient of Restitution
• Main factor in the formation of a tennis
racquet
• Ratio of rebound height to incident height
of the ball
• COR varies when the ball bounces off a
certain spot on the racket
• Maximum power is when COR is greatest
• COR=square root of Rebound height/initial
height
COR Data
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Center of Mass
1st set: hd = 20 cm, hr = 11 cm
2nd set: hd = 40 cm, hr = 22 cm
3rd set: hd = 60 cm, hr = 33 cm
4th set: hd = 80 cm, hr = 44 cm
5th set: hd = 100 cm, hr = 55 cm
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Tip of Head
1st set: hd = 20 cm, hr = 12 cm
2nd set: hd = 40 cm, hr = 24 cm
3rd set: hd = 60 cm, hr = 36 cm
4th set: hd = 80 cm, hr = 48 cm
5th set: hd = 100 cm, hr = 60 cm
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Near Throat
1st set: hd = 20 cm, hr = 8 cm
2nd set: hd = 40 cm, hr = 16 cm
3rd set: hd = 60 cm, hr = 24 cm
4th set: hd = 80 cm, hr = 32 cm
5th set: hd = 100 cm, hr = 40 cm
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COR
COR (center of mass): .74162
COR (near the throat): .63246
COR (tip of the head): .7746
hr = rebound height,
hd = Initial height
COR Data Analysis
• Rebound height
remains constant
• Best spot to hit the
ball
– Near tip
– Center of mass
– Disregard throat
The Dead Spot
• Spot near the tip where ball
doesn’t bounce at all
– All of the energy is given to
the racquet
– Racquet doesn’t give any
energy back to the ball
• Effective mass of the racquet
at that point is equal to the
ball
• Effective mass is F=ma,
therefore m=F/a
• When serving, the best place
to hit the ball is at the dead
spot
• When returning, dead spot is
the worst
Strings on the Racquet
• Act as a medium
– Absorb much of the ball’s kinetic energy
– Returns some of that energy back to the ball
• Tighter strings produce slower ball speeds
• Loose strings
– Leads to slightly higher rebound velocities
• More power
Conclusion
• Performance of the
racquet
– Strings
– Size of the head
• Larger the head = more
speed applied to ball
– Game depends on two
main things
• How well you hit with
the racquet
• The “spot”
Bibliography
• Brody, H. (1979) Physics of the tennis racket.
American Journal of Physics 47, 482-487
• Brody, H. (1981) Physics of the tennis racket II:
The sweet spot. American Journal of Physics,
49, 816-819.
• Brody, H. (1995) How would a physicist design a
tennis racket?. Physics Today, 48, 26-31
• Pallis, Jani. "The Flight of the Tennis Ball"
www.Tennisserver.com