🎾 Physics Of The Tennis Kick Serve¶
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Physics Of The Tennis Kick Serve — tài liệu 29 trang từ thư viện sách tennis.
Chủ đề chính: Topspin, Physics, Giao bóng
Tóm tắt nội dung (trích từ tài liệu gốc): Physics of the Tennis Kick Serve about:reader?url=http://twu.tennis-warehouse.com/learning_center/kicks... twu.tennis-warehouse.com Physics of the Tennis Kick Serve 38-48 ph�t ABSTRACT The kick serve in tennis is difficult to master since it is difficult to generate enough topspin for the ball to kick up sharply. Furthermore, the ball needs to be served at around 100 mph (depending on the court surface -- grass and clay being very different surfaces), and it needs to land well short of the service line in order to bounce to around shoulder height. The main problem is that the racquet is near t
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Physics of the Tennis Kick Serve
38-48 ph�t
ABSTRACT
The kick serve in tennis is difficult to master since it is difficult to
generate enough topspin for the ball to kick up sharply.
Furthermore, the ball needs to be served at around 100 mph
(depending on the court surface -- grass and clay being very
different surfaces), and it needs to land well short of the service
line in order to bounce to around shoulder height. The main
problem is that the racquet is near the top of its trajectory when it
strikes the ball, so it is impossible to swing up at the ball at the
same steep angle as that used in a topspin groundstroke. A high
ball toss will help since a falling ball is equivalent to a rising
racquet in terms of topspin generation. In addition, it helps to strike
the ball with the racquet head tilted forward slightly. Additional
topspin is generated simply by the fact that the racquet is rotating
forward when it strikes the ball. The physics of each of these
effects is described in this article, and is illustrated with slow
motion video film showing both the serve action and the fact that
the resulting spin is mostly sidespin in a typical kick serve.
1. INTRODUCTION
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A kick serve is one that bounces up around shoulder height as it
crosses the baseline. It also swings into or away from the receiver
due to sidespin imparted to the ball. Not everyone can serve a
good kick serve. The male pros all have a good kick serve, and so
do some of the female pros. The secret of success is being able to
serve with topspin (in addition to sidespin) and being able to serve
fast. In general, the faster the ball lands in the court the higher it
will bounce, but a fast serve on its own does not necessarily
bounce to around shoulder height. A fast, flat first serve usually
bounces to around waist height. In order to bounce to shoulder
height, the ball must land at high speed and at a relatively steep
angle.
Normally, the faster the serve, the lower the angle of incidence on
the court since a ball served at high speed needs to pass low over
the net in order to land in the serve box. But if the ball is served
with topspin, the ball will dive onto the opponent's court at a
relatively steep angle, even when serving at relatively high speed.
An interesting question is how the serve speed, the angle of
incidence of the ball onto the court and the amount of topspin can
all be optimized to generate the highest possible trajectory as the
ball crosses the baseline. In this article, I present some
measurements and calculations to indicate how the bounce height
can be increased. Measurements of kick serves were needed in
order to figure out the physics of the problem, which turned out to
be more complicated than expected.
One of the mysteries concerning the kick serve is how the server
manages to generate topspin in the first place. In order to achieve
a good kick serve, the racquet head needs to rise up the back of
the ball, as it does in a topspin groundstroke. To return a
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groundstroke with topspin, players normally swing the racquet
upwards at an angle of about 30 degrees to the horizontal. That
way, an upwards friction force is exerted on the back of the ball,
causing the ball to rotate with topspin if the racquet is swung fast
enough. The incoming ball bounces off the court with topspin, so
the spin direction needs to be reversed in order to return the ball
with topspin. In a serve the racquet head is almost at the top of its
trajectory when it strikes the ball, so the racquet is rising at an
angle of only a few degrees just before it strikes the ball. Therein
lies the mystery. If the racquet needs to rise at 30 degrees to hit a
good topspin forehand, how can anyone serve a ball with a
significant amount of topspin when the racquet head is rising at
only a few degrees?
One saving grace is that the ball is not spinning backwards when it
is struck, so the server does not need to apply as much spin in a
kick serve as in a groundstroke. In effect, the same outgoing spin
can be achieved with only about half the effort. Another significant
factor is that the racquet is usually swung faster when serving a
ball than when hitting a groundstroke. The outgoing spin is
proportional to the speed of the racquet head, and is also
proportional to the approach angle of the racquet head. The first
two factors help to increase the amount of topspin in a kick serve,
but there are several other factors that also add to the spin, as
described in Section 2.
When serving a kick serve, right-handed players toss the ball over
their left shoulder, arch their back, bend at the knees and then
jump up off the court. The end result is that the racquet head
strikes the ball in a direction that is partly sideways across the
back of the ball and partly vertical up the back of the ball, as
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shown in Fig. 1. The sideways component generates sidespin and
the vertical component generates topspin. In Fig. 1, the ball will
have more sidespin than topspin since the horizontal speed of the
racquet head is greater than the vertical speed. Sidespin causes
the ball to curve from right to left through the air, as viewed by a
right-handed server, or from left to right for a left-hander. Topspin
causes the ball to curve down onto the court at rate that is faster
than the effect of gravity alone.
Figure 1 -- Direction of racquet head and spin axis in a kick serve,
as viewed by (a) a right hander and (b) a left hander. The ball is
traveling into the page toward the net. The spin is primarily
sidespin here with a small topspin component. The aerodynamic
Magnus force, F, acts at right angles to the spin axis, pushing the
ball down onto the court and causing it to curve to the left in (a) or
to the right in (b). F is in the opposite direction to the friction force
on the ball generated by string motion across the back of the ball.
2. SPIN GENERATION
The situation shown in Figure 1 is the one normally used to
describe how players strike the ball in a kick serve, and it shows
how sidespin is generated as well as topspin. That doesn't mean
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that the ball spins about two separate axes. There is only one spin
axis and it is tilted away from the vertical. If the axis in Fig. 1 is
vertical, then there is no topspin, just sidespin. If the axis is
horizontal, then there is no sidespin, just topspin. If the axis is
tilted, then any point on the ball will rotate in a circle around the
axis, and it rotates simultaneously in vertical and horizontal
directions.
Video film of a kick serve shows that the situation is more
complicated than that shown in Fig. 1. There are several additional
effects that need to be considered in order to understand what
happens in a kick serve. The main effects are these:
1. Racquet Tilt. If the racquet head is tilted forward when the head
strikes the ball, rather than being exactly vertical, then additional
topspin is generated and the ball will pass lower over the net. The
same effect occurs in a topspin groundstroke. The effect of racquet
head tilt is shown in Fig. 2. Suppose that the head is tilted forward
and approaches the ball in a horizontal direction at 80 mph, as in
Fig. 2(a). The physics of the collision is exactly the same if the
racquet is at rest and the ball approaches at 80 mph as in Fig.
2(b). That is what a bug would see if the bug was sitting at rest on
the frame of the racquet. Since the ball approaches the racquet at
an angle, it will bounce off the racquet at an angle with topspin.
The same thing would happen if the ball bounced off the court at
an angle. The result of the collision in Fig. 2(a) is that the ball is
served in a downward direction with topspin, even if the racquet
head is not rising when it strikes the ball.
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6 trong 29 Figure 2 -- (a) If a racquet approaches a ball at 80 mph with the
head tilted forward, then the ball will be served with topspin even if
the head is not rising. The same amount of spin is generated if the
ball approaches the racquet at 80 mph and the racquet is initially
at rest, as shown in (b).
2. Gyrospin. Some players tend to strike the ball slightly towards
the side of the ball rather than exactly across the back of the ball,
in which case the ball will also spin about a horizontal axis pointing
towards the net. That sort of spin is used to throw a gridiron
football, but it has no effect on curvature of the ball through the air.
3. Ball toss. If the ball is struck when the racquet is exactly vertical
and is at its maximum height, then there is no vertical motion of
the racquet head at all. Nevertheless, the ball will still be struck
with topspin, and it will head downward towards the net rather than
being served in a horizontal direction. Effects 3 , 4 and 5 can all
lead to this result and they all result in additional topspin when the
head is rising to meet the ball.
When the ball is struck, the ball is falling down towards the court
as a result of the ball toss. If the ball falls down onto the racquet
strings, then that is equivalent to the racquet head rising to meet
the ball. The faster the head rises and the faster the ball falls, the
more topspin is generated. If the head is rising and the ball is not
falling then the ball will be struck in an upwards direction. If the
head is not rising but the ball is falling, then the ball will be struck
in a downwards direction. Normally, the head is rising and the ball
is falling when the ball is struck, and the result is that the ball
usually heads toward the net in a downward direction -- partly
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because of the ball toss effect and partly because of effects 1, 4
and 5.
4. Downward force. The racquet head is rotating rapidly towards
the net when the ball is struck. The racquet head rotates through
an angle of about 10 degrees while the ball is on the strings. If the
ball is struck when the head is vertical, then the ball will come off
the strings when the head is tilted forward by about 10 degrees.
The ball will come off the strings at an angle of about five degrees
below the horizontal, rather than 10 degrees, since the force on
the ball is a maximum half way through the impact, and is zero at
the start and end of the impact. On average, the force on the ball
acts in a direction about 5 degrees below the horizontal in that
case. The serve angle also depends on the height of the ball toss
and whether the head is rising when it strikes the ball. The serve
angle is very important in a fast serve since an error of two
degrees can result in a fault where the ball either hits the net or
lands long.
5. Racquet Rotation. The strings grip the ball during a serve. If the
racquet rotates 10 degrees while the ball is on the strings, then the
ball also rotates 10 degrees, in the topspin direction. The same
effect would occur if the ball was glued to the strings since the ball
and the racquet would both rotate 10 degrees. The ball is not
glued to the strings but it is squashed against the strings. The top
end of the racquet is rotating faster than the bottom end, so the top
side of the ball is pushed harder towards the net than the bottom
side. As a result, the ball rotates with topspin, even if the racquet
head is not rising when it strikes the ball. If the racquet head is
indeed rising when the ball is struck, then the amount of topspin
will increase. Ten degrees of forward rotation in 0.004 seconds
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corresponds to about 400 rpm of topspin, which is about the same
as one gets when the racquet head is rising. In a kick serve, the
ball spins at around 4000 rpm, but most of that is due to sidespin.
The amount of topspin is typically less than 1000 rpm.
Figure 3 shows a simple model of the rotating racquet effect. The
ball can be represented by a mass M plus two springs. If the
racquet moves in a straight line towards the ball, both springs
compress equally and the ball accelerates in a straight line without
rotating. If the racquet is rotating then the top spring compresses
more than the bottom spring so the force N1 is greater than N2
and the ball will rotate in the same direction as the racquet. If the
racquet head is also rising, then the back of the ball grips the
strings and rises with the racquet while the front tends to remain at
rest, and this will generate additional topspin. If the ball is falling
when it is struck, the back of the ball is gripped by the strings while
the front keeps falling, adding to the amount of topspin.
Figure 3 -- A ball can be represented by a mass M and two
springs. If the racquet rotates then so does the ball since the top
spring will compress more than the bottom spring and will exert a
greater force on the ball.
3. TYPICAL SERVE RESULTS
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Video film showing a few kick serves can be seen below in Movie
Screen 1. Film was taken at 300 fps (frames per second) looking
side-on (along the baseline), then the camera was moved to view
from the rear (looking along the center line). The camera was then
moved back to the side-on position to film at 600 fps while zoomed
in close to see the ball more clearly (Figure 4).
Figure 4 -- Camera positions used to film kick serves.
The ball was marked with various dots, circles and lines so that
both the spin rate and the spin axis could be determined. In
addition, measurements were made from the video film of the
speed and angle of the racquet head, as well as the speed and
angle of the ball. A 25 fps camera was also positioned on the other
side of the net to determine the landing position of the ball, as well
as the incident and rebound speeds and angles. Eight players
were selected from high ranked juniors coached by Tennis NSW,
with an average age of about 20, and filmed in March 2011 during
a regular coaching session.
Choose Movie:
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0:00 / 0:14
Movie Screen 1. -- Side View (300 fps). Racquet head is still rising
for a few frames even after the ball comes off the strings. Even
though the ball is struck while the head is rising, the ball travels
downward toward the net because of the high ball toss and
rotation of the racquet during the impact. The high ball toss means
that the the ball drops rapidly both before and after the impact.
Rear View (300 fps). This serve kicked up well, mainly because it
landed well short of the serve line. The ball bounced to its
maximum height near the back fence.
Impact 1 (600 fps). Racquet head is still rising for a few frames
even after the ball comes off the strings. The ball appears to have
pure topspin at one stage but the spin is mainly sidespin. The axis
is tilted about 30 degrees away from the vertical giving some
topspin as well as sidespin. The pattern on the ball repeats every
11 frames so the ball is spinning at 3270 rpm.
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Impact 2 (600 fps). Racquet head has almost reached the top of its
swing and is rising only a few degrees on impact. The ball heads
off a few degrees below the horizontal with a small amount of
topspin, but the spin is mainly sidespin. The pattern on the ball
repeats every 8 frames, so spin = 4500 rpm.
Impact 3 (600 fps). Essentially the same serve as Impact 2,
although it appears that the ball has no topspin at all. In fact, the
axis is tilted about 10 degrees giving a small amount of topspin.
The equator line is perpendicular to the axis and is also tilted
about 10 degrees below the horizontal so it does not rotate in the
manner that one would normally expect for a ball struck with
topspin. Ball pattern repeats every 10 frames so spin = 3600 rpm.
Flat/Kick Serve Rear View Comparisons (300 fps). These videos
illustrate the difference between a first serve (top video) and a kick
serve (bottom video) for the same player. The ball kicks up higher
in the kick serve, even though the ball lands near the same spot
lengthwise, about 5 feet from the serve line. The ball toss position
is different and the sideways motion of the racquet head across
the ball is much different. In the flat serve, the ball takes 128
frames to land on the court (128/300 = 0.416 sec). In the kick
serve the ball lands after 156 frames (t = 156/300 = 0.520 sec).
Flat/Kick Serve Side View Comparisons (300 fps). Both serves
start out exactly the same (top-flat; bottom-kick), and even the ball
toss looks the same. It is the sideways motion of the racquet that
is different, not the forward or upward motion.
(Note: To see spin and angles at impact, movies are best viewed
frame-by-frame using keyboard arrow keys or movie controls.)
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4. INTERPRETATION OF SPIN
Measurement of spin rates from high speed video film is relatively
easy. The spin axis remains fixed while the ball travels toward the
net, so the time for one revolution can be measured in terms of the
number of video frames required for a particular mark or pattern on
the ball to re-appear in the same orientation. When filming at 600
fps, the mark re-appeared after about 8-10 frames or about 8/600
to 10/600 seconds, giving a spin rate of 60 to 75 revolutions/sec or
3600 rpm to 4500 rpm.
It is much more difficult to locate the spin axis, unless the spin axis
happens to be exactly vertical or exactly horizontal. If the spin axis
is vertical, then all marks on the ball rotate in a horizontal direction,
and a horizontal line around the equator remains horizontal, as
indicated in Fig. 5(a). If the spin axis is horizontal and points to the
camera, then all marks on the ball rotate in a circular path around
the middle of the ball. Otherwise, the marks and lines rotate in a
manner that can be difficult to interpret. For example, Figs. 5(b)
and (c) show two positions of the equator, one half revolution
apart, when the axis is vertical and the equator line is inclined at
an angle to the axis. It might appear that the ball is rotating with
topspin, given the rotation of the equator line during half a
revolution, but if the axis is vertical then there is no topspin at all.
In that case, marks on the ball rotate purely in the horizontal
direction and then disappear around the back of the ball.
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Figure 5 -- Lines and marks on a ball can be used to determine
the location of the spin axis. The red dot here marks the position
where the axis passes through the surface of the ball.
Figure 5(d) shows a situation where the axis is perpendicular to
the equator line but the top end of the axis is tilted to the left and is
also tilted out of the page. In that case, the equator line appears
stationary on video film, giving the false impression that there is no
topspin. All marks on the ball rotate in circular paths around the
axis, so the axis can be identified by the motion of those marks. If
the axis is horizontal and pointing in the same direction as the
motion of the ball, then the spin is classified as gyrospin. That type
of spin is used when throwing an oval-shaped football. If the spin
axis is essentially vertical but tilted in a direction toward the net in
a serve, then the ball will have a small gyrospin component. If the
spin axis is tilted sideways, in a direction parallel to the net or the
baseline (as in Fig. 1) then the ball will have a small topspin
component.
The approach used by the author to determine the spin axis was to
mount a ball in such a way that its axis could be fixed in any given
position, and then to rotate the ball about that axis in order to
compare the result with the video film. A certain amount of trial and
error was needed to identify the spin axis, but it was usually close
to the orientation shown in Fig. 1. That is, the axis was usually
tilted away from the vertical by about 10 or 20 degrees, although it
was also tilted slightly toward the net in some cases, meaning that
the ball was struck slightly toward the front of the ball rather than
exactly at the rear of the ball.
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5. ESTIMATION OF TOPSPIN
Figure 6 -- (a) Racquet head approaches the ball at speed V and
angle A. The ball is falling slowly. (b) The ball emerges at speed v
and with topspin, at S rpm. In a kick serve, the angle A is only a
few degrees, but is shown here as a relatively large angle for
clarity.
The amount of topspin generated in a serve due to vertical motion
of the racquet head can be estimated by considering the situation
shown in Fig. 6. The racquet head is vertical and is approaching
the ball rapidly at speed V and angle A, while the ball falls slowly.
We can ignore sideways motion of the racquet head for the
moment, in order to calculate the amount of topspin. Sideways
motion is a separate issue and it generates sidespin, in the same
way that vertical motion of the racquet head generates topspin.
After the racquet head strikes the ball, the ball will emerge at
speed v and with topspin, as indicated in Fig. 6(b). The ball
emerges at high speed towards the net and it usually heads
slightly downward toward the net. In Fig. 6(b), the ball is shown
heading slightly upward since that is the effect of the upward
friction force of the strings acting on the back of the ball. The
friction force must act in an upward direction to generate topspin.
If the ball is struck when the strings are vertical, and if the head is
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moving in a horizontal direction without any rotation when it strikes
the ball, then there will be no upward force on the ball. The only
force on the ball in that case would be a horizontal force. There
would be no change in the vertical speed of the ball, so the ball
would continue to fall vertically at the same vertical speed it had
just before it was struck. The ball would then head downward at a
few degrees to the horizontal.
If the ball is struck when the strings are vertical, and if the racquet
head is rising when it strikes the ball, then there is a vertical force
on the ball that causes it to rotate with topspin and that reduces
the speed at which the ball falls as a result of the ball toss.
Furthermore, the racquet head tends to be inclined forward slightly
when it strikes the ball, which generates an additional downward
force on the ball. In that case, the whole diagram in Fig. 6 needs to
be rotated slightly so that the racquet head is tilted forward and the
ball emerges in a downward direction. In addition, the racquet
head rotates a few degrees while the ball is still on the strings,
which also helps to project the ball downward.
The amount of topspin is shown in Fig. 6 with the symbol S.
Experiments and theoretical estimates both indicate that S is given
to a good approximation by S = 1.45 VA where S is the spin in
rpm, V is the racquet head speed in mph and A is the approach
angle in degrees. For example, if A= 0 then S = 0 meaning that
there is no spin generated at all. If V = 100 mph and A = 30
degrees then S = 4350 rpm. The amount of spin therefore
increases with both the speed of the racquet head and the
approach angle of the racquet head. The amount of spin also
depends on the speed of the incoming ball, but in a serve, the ball
is almost at rest when it is served.
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Figure 6 also describes the result when the racquet is moving
sideways across the back of the ball, and is approaching the ball
at a sideways angle A. In that case, the ball acquires sidespin, and
the amount of sidespin is given by the same expression S
=1.45VA. In practice, the racquet head usually approaches the ball
as shown in Fig. 1, with a large sideways approach angle and a
relatively small vertical approach angle. As a result, the ball is
usually served with about 4000 rpm of spin in a kick serve, but the
spin is mostly sidespin and the amount of topspin is relatively
small. That is, the spin axis is almost vertical, as indicated in Fig.
1.
In a typical kick serve, the racquet head approaches the ball at
about 65 mph and rises at an angle of about 5� just before
impacting the ball. The amount of topspin in that case is about S =
1.45 � 65 � 5 = 471 rpm. If the ball toss is not right, and the ball is
struck a bit further forward, the vertical approach angle of the
racquet head might be only one degree, then the amount of
topspin will be five times smaller. If the ball is struck a bit earlier,
the approach angle might be 10� instead of 5� then the amount of
topspin will double. However, if the ball is struck too early, then it
might land on the baseline instead of the service line. Hitting up at
a greater approach angle to the ball generates more topspin, but
the ball is then launched at a higher angle over the net.
6. SPIN DUE TO BALL TOSS
Suppose a racquet approaches a ball in a horizontal direction at
speed V and the ball is falling vertically at speed v as shown in Fig.
7(a). We can work out the amount of topspin by supposing that the
ball is at rest and the racquet is rising vertically at speed v while
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simultaneously moving horizontally at speed V, as shown in Fig.
7(b). The situation is then the same as that shown in Fig. 6 and the
spin will be given by the same formula. If the ball falls say 3 feet
before it is struck then it will be falling at 9.5 mph when it is struck.
If it falls from a greater height it will be falling at a greater speed
when it is struck. In a kick serve, V is typically about 65 mph if the
ball is served at 90 mph. The racquet approaches the ball at an
angle A given by tan A = v/V = 0.146 in this case, so A = 8.3�. That
angle could well be larger than the actual vertical approach angle
of the racquet in a typical kick serve, in which case the ball toss
would account for more than half of the topspin generated. Using
the formula S = 1.45VA, we find that S = 780 rpm due to the ball
toss alone.
Figure 7 -- (a) A racquet approaches a ball at speed V while the
ball is falling at speed v. A bug on the ball sees the ball at rest
while the racquet is rising at speed v, as shown in (b).
7. BALL TRAJECTORY AFTER BOUNCING
In the remainder of this article, I present some theoretical
calculations of ball trajectories in a kick serve. The calculations are
based on experimental observations, but it is easier to show the
effects of varying ball spin, speed and angle by calculating the
effects rather than by measuring them. The calculations are
presented in two stages. First, we examine the ball trajectory after
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the ball bounces, since that is what determines the height of the
ball as it crosses the baseline. Then we examine the whole
trajectory from the serve point to the baseline to show how bounce
height is affected by changes in the serve action.
Figure 8 -- The ball crosses the baseline at height H after
bouncing up off the court at speed v and angle . In a kick serve, v
is about 20 m/s, is about 20� and H is about 5 or 6 ft. The ball
bounces with topspin, at about 600 rad/s (5730 rpm). The distance
from the bounce point to the baseline is about 6 to 8 m (20 to 26
ft). The perpendicular distance from the net to the service line is 21
ft. The perpendicular distance from the service line to the baseline
is 18 ft.
After the ball lands in the service box, it bounces up off the court at
speed v, at an angle , with topspin , and then crosses the
baseline at height H, as shown in Fig. 8. In general, the height H
increases as v increases, it increases as increases, and it
decreases as increases. Topspin causes the ball to dive down
onto the court which is good if you want the ball to land at a steep
angle in the service box and to kick up at a steep angle. After the
ball bounces, the effect of topspin is to reduce the bounce height.
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Figure 9 -- Height of the ball as it crosses the baseline, either 6.3
m or 8.0 m from the bounce point.
Effects of varying these parameters are shown in Fig. 9, based on
trajectory calculations that allow for aerodynamic lift and drag
forces on the ball. Fig. 9(a) shows the effect of changing the
bounce speed, assuming that the ball bounces at = 20� with 600
rad/s (5370 rpm) of topspin. The ball then travels a horizontal
distance of either 6.3 m (20.7 ft) or 8.0 m before crossing the
baseline. The shortest distance from the service line to the
baseline is 5.49 m (18.0 ft). The ball needs to travel a longer
distance to the baseline if it bounces before reaching the service
line, especially when served wide rather than down the middle.
Fig. 9(b) shows the effect of varying the bounce angle, and Fig.
9(c) shows the effect of varying the rebound spin of the ball. The
ball speeds, spins and angles shown in Fig. 9 are all typical of
those in a kick serve. Fast, flat serves bounce off the court at an
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angle of about 14�.
The bounce speed and angle both have a big effect on the bounce
height. The higher the bounce angle the better, and the higher the
bounce speed the better. The distance to the baseline is also
important, given that the ball is still rising as it crosses the baseline
in most of the cases in Fig. 9. If the ball bounces well short of the
serve line then it has further to travel to the baseline so it will cross
the baseline at a greater height. The ball always bounces with
topspin. The amount of topspin after the bounce increases with
serve speed and with the angle of incidence onto the court, but the
amount of topspin does not have as strong an effect on the bounce
height as the bounce speed or angle.
It is clear from these calculations that a good kick serve is one
where the ball bounces off the court at a large angle and at high
speed, especially if the ball lands in the service box well short of
the baseline. An alternative method of getting the ball to bounce
over the receiver's head is available at low serve speeds, although
no one ever uses this method. Figure 10 shows the height of the
ball as it crosses the baseline, for low bounce speeds and high
bounce angles. These results could be obtained by serving
underarm or overarm so the ball falls from a large height into the
service box and bounces to a large height. Alternatively, the ball
could be smashed or volleyed at low speed into the service court
to bounce in this manner. In order to calculate the results in Fig. 10
it was assumed that the ball spin changes with bounce speed and
angle in such a way that the ball bounces in a rolling mode, as it
normally does when incident on the court at a large angle of
incidence.
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Figure 10 -- Height of the ball as it crosses the baseline, either 6.3
m or 8.0 m from the bounce point.
8. SERVE PARAMETERS
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Figure 11 -- Height of the ball as it crosses the baseline, vs serve
angle A in (a) and serve speed V in (b), (c) and (d). The angle A is
the serve angle in degrees below the horizontal. The ball spin is
either 3000, 4000 or 5000 rpm, as labeled, and the spin axis is
tilted at either 20� or 30�, as labelled. A larger tilt angle means
more topspin. If the serve angle is too small or the serve speed is
too high, the ball will land long (beyond the serve line). If the serve
angle is too large or the serve speed is too low, the ball will not
clear the net. It was assumed that the ball is served down the
center line for these calculations.
In order to get the ball to bounce up off the court at high speed and
at a large angle, the ball must normally be served at high speed
and with topspin. Calculations for a range of serve speeds, serve
spins and serve angles are shown in Fig. 11. The spin values
shown in Fig. 11 refer to the amount of spin generated as the ball
comes off the racquet, not the spin after the ball bounces off the
court. Despite the fact that the server swings up at the ball in a
kick serve, the ball must be projected downward below the
horizontal for a good serve. The serve angle, A, is typically
between 2 and 6 degrees below the horizontal. It was assumed in
Fig. 11 that the ball is served down the center line when deciding
whether the ball hit the net or was long, and when calculating the
height of the ball as it crossed the baseline.
The height of a ball served at 80, 90 or 100 mph as it crosses the
baseline is shown as a function of the serve angle in Fig. 11(a),
assuming that the ball is spinning at 4000 rpm and the spin axis is
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tilted 30� from the vertical. It is clear from this diagram, and the
other results in Fig. 11, that the ball needs to be served at a speed
greater than 80 mph for a good kick serve. At 80 mph, the ball
bounces to a height of just over 4.5 feet, regardless of the spin and
serve angle. The highest bounces occur when the ball is served at
about 90 to 100 mph and at about 6� below the horizontal so the
ball just clears the net.
Figure 11(b) shows the bounce height at the baseline vs serve
speed when A = 4�, for three different values of ball spin. The
bounce height increases with spin, but it does not necessarily
increase with serve speed. At high serve speeds the ball lands
closer to the serve line, assuming the serve angle is fixed at 4�, so
the angle of incidence is low and the ball bounces at a low angle.
In order to increase the bounce height, the ball needs to be served
at a steeper angle, as shown in Fig. 11(c). However, if the amount
of topspin is reduced then the ball will again land close to the
service line and the bounce height is reduced, as shown in Fig.
11(d).
Several assumptions were made in calculating the results shown
in Fig. 11. The distance between the two baselines is 78 feet, but
the distance between the server and the point at which the ball
crosses the opposite baseline can be greater than 78 feet,
especially when serving wide. To calculate the results in Fig. 11 it
was assumed that the ball is served down the center line. In that
case, the relevant distance to the opposite baseline is 78 ft. Larger
bounce heights can be obtained by serving wide rather than down
the center line. The ball was served from a height of 2.9 m (9.5 ft),
starting 0.6 m (2 ft) in front of the server's baseline. To calculate
the change in ball speed when the ball bounced, it was assumed
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that the vertical speed after the bounce was 0.8 x the vertical
speed before the bounce (i.e., COR = 0.8), and it was assumed
that the horizontal speed, vx2, after the bounce was 0.65 x the
horizontal speed before the bounce. The calculations were slightly
simplified by assuming that the ball bounces in a rolling mode, with
vx2 = R2, where R is the ball radius and 2 is the angular
velocity of the ball (in rad/s) after it bounces. The bounce
parameters here are typical values, chosen from experimental
data.
The specific parameters used to calculate the results in Fig. 11 are
of less significance than the general trends, which show that the
bounce height at the baseline increases with the amount of topspin
imparted by the server, and it also increases as the serve angle, A,
increases. Serving downward at a relatively large angle has two
main advantages. It means that the angle of incidence on the court
will be large, so the bounce angle will be large. It also means that
the ball lands well before the service line, so the ball has a longer
distance to travel before it crosses the baseline. At moderately
high bounce speeds, the ball rises all the way 15 to the baseline,
so the longer travel distance allows the ball to rise to a greater
height by the time it reaches the baseline. The advantage of
serving with topspin is that the ball is incident on the court at an
even steeper angle and at a greater vertical speed than a ball
served without topspin.
9. GENERATION OF TOPSPIN
It is interesting to consider how topspin is generated when viewed
from the perspective of a moving ball striking a stationary racquet.
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Even though the ball is almost at rest when it is struck in a serve, it
is useful to consider the serve in a reference frame where the
racquet is at rest just before the impact with the ball. If a racquet is
swung at 80 mph at a stationary ball, a bug sitting on the racquet
strings might think it was at rest and the ball was approaching at
80 mph, as shown in Fig. 12(b).
Figure 12 --A racquet approaching a ball at 80 mph, as in (a), can
be viewed as if the ball approaches the racquet at 80 mph, as in
(b). The ball will bounce off the racquet at about 25 mph, but in the
court frame of reference (c) the ball is served at about 100 mph.
A ball incident on a stationary racquet at 80 mph will bounce off
the racquet at about 25 mph and with topspin. Viewed in the court
frame of reference, as in (c), the ball is served at about 100 mph.
The actual speed, spin and angle of the served ball depends on
the approach angle of the racquet, the tilt angle of the racquet
head and the speed of approach of the racquet head. If the
racquet head was vertical at impact and approached the ball in a
horizontal direction, then the ball would be served in a horizontal
direction without any topspin. In order to increase the amount of
topspin, the angle of incidence of the ball onto the strings, as
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shown in Fig. 12(b), needs to be increased, a result that can be
achieved by tilting the racquet head forward and by increasing the
approach angle of the racquet head. Tilting the racquet head
forward, as indicated in Fig. 12, also has the effect of projecting
the ball down below the horizontal.
10. TOPSPIN VS SIDESPIN
Suppose that a ball is spinning at 4000 rpm and the spin axis is
tilted 30� away from the vertical. The spin of the ball is completely
specified by these numbers, but does it make sense to say that the
ball has say 3000 rpm of sidespin and 1000 rpm of topspin? Can
we add the separate spins in this way to calculate the total spin?
In order to calculate the results in Fig. 11, I did not separate the
spin into topspin and sidespin in order to calculate the force on the
ball. Rather, I worked out the Magnus force (F in Fig. 1) from the
"total" spin and then calculated the vertical component of that
force, FV, knowing the tilt angle. If FM is the Magnus force on the
ball and is the tilt angle, and if the ball is traveling in the
horizontal direction, then FV = FM sin. For example, if the spin
axis is tilted by 30� then FV = 0.5FM so the vertical force on the
ball due to spin is half the total force. The horizontal force due to
the ball spin is given by FM cos = 0.87FM. The sideways force on
the ball is reduced only slightly when the spin axis is tilted by 30�
but the vertical force increases by a relatively large amount. The
two forces don't simply add up in the usual way to give the total
force since forces add as vectors.
Similarly, the two spin components don't add up in the usual way
to give the total spin. Nevertheless, spin can be regarded as a
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vector with magnitude and direction and can be regarded as
having vertical and horizontal components or separate sidespin
and topspin components. But it is difficult to then calculate the
Magnus force on the ball from the two separate components, the
problem being that the aerodynamic force on the ball depends on
the total spin of the ball, not just the topspin or the sidespin
component. The vertical force on the ball cannot be calculated just
from the topspin component. The amount of sidespin also needs to
be taken into account.
11. ANSWERS TO QUESTIONS
Q: Can a kick serve be hit slower than 80 mph?
A: Yes. The graphs above in Figure 11 refer specifically to the
height of the ball as it crosses the baseline when the ball is served
down the center line. In most cases, the ball is still climbing as it
crosses the baseline, so the eventual height of the ball might
exceed 6 ft. It is also possible to serve a topspin lob at low speeds
with a high bounce, as shown in Figure 13 below, but the ball then
drops sharply as it crosses the baseline. Even higher bounces can
be expected on clay courts since the ball slows down more in the
horizontal direction and therefore kicks up at a steeper angle.
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Figure 13 -- Low Speed, High Top Spin 'Lob Kick Serve'. The ball
leaves the racquet at an upward angle, bounces high, but then
drops sharply as it crosses the baseline.
Q: Does the ball toss really contribute to spin?
A: Yes. Movie Screen 2 demonstrates this. The pendulum in the
videos is an upside-down version of a serve -- a rising ball in the
video is equivalent to a falling ball in a serve. As the videos show,
in a serve, if the ball toss is falling, topspin is created. If the ball is
rising, underspin is the result. If the ball is stationary at its peak,
then there is no spin. These results are for a horizontally moving
pendulum (racquet). The pendulum speed is too slow and rotation
contact too small (1-2 degrees) in these videos to demonstrate the
racquet rotation effect discussed in Section 2.
Choose Movie:
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0:00 / 0:05
Movie Screen 2. -- Effect of Ball Toss on Spin.
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