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378 RESONANCE April 2008
Classroom
In this section of Resonance, we invite readers to pose questions likely to be raised in a
classroom situation. We may suggest strategies for dealing with them, or invite responses,
or both. “Classroom” is equally a forum for raising broader issues and sharing personal
experiences and viewpoints on matters related to teaching and learning science.
Spinning Ball Flight Under Moderate WindK R Y Simha
Mechanical Engineering
Department
Indian Institute of Science
Bangalore 560 012, India.
Email:
Dhruv C Hoysall
Indian Institute of Technology
Madras
Chennai 600 013, India.
Email:
Keywords
Magnus effect; cycloid, spiroid,
terminal velocity, sine/cosine
integral, leg spin, off spin, s-
spin, top spin, side spin, back-
ward spin.
P r e d ic t in g t h e ° ig h t o f s p in n in g b a lls is a n e x -
c it in g a s p e c t w h ile p la y in g c r ic k e t , t e n n is , t a b le
t e n n is o r s o c c e r . T h e s e g a m e s d e m a n d a w id e
r a n g e o f s k ills t o e x p lo it t h e a e r o d y n a m ic e ® e c t s
in d u c e d b y s p in , w in d a n d g r a v it y . T h e t h e -
o r y u n d e r ly in g t h e s e a e r o d y n a m ic e ® e c t s u n v e ils
b iz a r r e o p p o r t u n it ie s fo r b u d d in g c r ic k e t , t e n n is
a n d s o c c e r s t a r s !
1 . I n t r o d u c t io n
A p re v io u s c la ssro o m a rticle [1 ] d iscu sse d th e e ® e c t o fstro n g w in d o n ten n is b a ll ° ig h t. H o w ev e r, m o re c o m -m o n ly, m o d e ra te w in d s p re v a il d u rin g p la y tim e . A lsob a d sh o ts h it b y b a tsm e n o ® th e e d g e o f th e b a t in d u cesp in to th e b a ll. T h e c o m b in e d e ® e c t o f sp in , d ra g a n d
g ra v ity p ro d u c e sp e cta cu la r e ® e c ts in te n n is b a ll c rick e ta n d so c ce r. A s d isc u sse d in [1 ], th e g o v e rn in g d i® e re n tia le q u a tio n s o f m o tio n u n d e r m o d e ra te w in d c o n d itio n a rea fo rm id a b le set o f co u p le d n o n -lin e a r d i® e re n tia l e q u a -tio n s! In g e n e ra l, th e n o n lin ea r e q u a tio n s d e fy a n a ly tic a l
stra te g ie s a n d d e m a n d n u m erica l m e th o d s fo r th eir so -lu tio n s. In th is a rtic le, w e re v isit th ese e q u a tio n s, a n d
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379RESONANCE April 2008
re -ex a m in e th e sp e cia l c a se o f th e b a ll re tu rn in g to th ep o in t o f p ro je ctio n . F o r th e sa k e o f c o n tin u ity, w e re -c a p tu la te th e m a in id ea s fro m th e p rev io u s c la ssro o m
a rticle o n ten n is b a ll ° ig h t w ith o u t sp in u n d er stro n gw in d .
2 . F lig h t W it h o u t S p in : S t r o n g W in d s
S u p p o se a b a tsm a n h its a ten n is b a ll. L et th e b a ll v e -lo c ity b e v b g , w h e re su p e rsc rip ts d en o te th e v e lo c ity o fth e b a ll w ith re sp e c t to th e g ro u n d . A ssu m in g a ste a d yw in d v e lo c ity v w g , th e rela tiv e v e lo c ity o f th e b a ll w ith
re sp e c t to th e w in d is v b w = v b g ¡ v w g . In g e n era l, v b g
a n d v b w h a v e a ll th re e c o m p o n e n ts w h ile v w g h a s o n lytw o co m p o n e n ts p a ra lle l to th e c rick e t g ro u n d . T h e d ra gfo rc e v a rie s a s th e sq u a re o f th e re la tiv e v e lo c ity m a g -n itu d e (v b w )2 a n d a c ts in th e d ire c tio n o p p o site to th e
v e c to r v b w . In v ec to r n o ta tio n , th e d ra g ex p re sse d p e ru n it m a ss is
D
m= ¡ k (v b w )2 v b w
v b w= ¡ k v b w v b w : (1 )
In e q u a tio n (1 ) k is th e d ra g c o e ± c ie n t, w h ich d ep e n d so n th e sh a p e , siz e , o rie n ta tio n , sp e ed a n d te x tu re (fu r,m o istu re, e tc.,) o f th e ° y in g o b je ct. It is c o n v e n ien t toin tro d u c e a sta n d a rd w in d sp ee d c o , w h ich p ro d u ce s ad ra g eq u a l to th e b a ll w e ig h t. F o r a sta n d a rd 6 0 g te n n isb a ll, c o is a b o u t 2 5 m / s. T h u s,
D = ¡ m gv b w
c 2o
v b w :
W e a re n o w w e ll a rm e d to a tta ck th e v e cto r eq u a tio n o fm o tio n :
¡ m a b g = (D + m g ); (2 )
w h e re a b g is th e a c c ele ra tio n v e c to r. In o rd e r to ¯ x th ed irec tio n s o f th e u n it v e cto rs, w e ta k e th e b a tsm a n a sth e o rig in o f c o -o rd in a tes (X ,Y ,Z ). T h e X -a x is is a lo n g
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380 RESONANCE April 2008
th e p itch , Y -a x is is p o in tin g in th e d ire c tio n o f th e le gu m p ire , a n d Z -a x is p o in tin g to th e sk y a b o v e th e b a ts-m a n . T h e w in d v e cto r is v w g = (iv w g
x + jv w gy ) a n d th e
a c c ele ra tio n d u e to g ra v ity is { k g . T h e v ec to r e q u a tio no f m o tio n lo o k s d ec e p tiv ely sh o rt a n d sim p le , b u t w a itu n til y o u rea d a b o u t a ll th e c o m p lic a tio n s ca u se d b y th eD te rm ! E x p a n d in g th is eq u a tio n in to its sc a la r c o m p o -n e n ts.
a b gx = ¡ g
v b wx v b w
c 20
;
a b gy = ¡ g
v b wy v b w
c 20
;
a b gz = ¡ g
"
1 +v b w
z v b w
c 20
#
: (3 )
C lea rly , th e re is n o h o p e fo r a n e a sy so lu tio n c o n sid e rin gth a t h a rd ly a n y th in g is k n o w n a b o u t th e v a ria tio n o f a b g
a n d D w ith tim e in th e th re e-d im e n sio n a l, w in d -b lo w nsp a ce . It se e m s lik e it is n o t g o o d c rick e t! B u t, w a it!
F o r stro n g w in d s v b w ¼ ¡ v w g ! T en n is b a ll c rick e t d o e sn o t sto p b ec a u se o f a g a le o r tw o ! N o w , u n d e r th is g a lefo rc e , th e fo rm id a b le n o n lin ea r c o u p led e q u a tio n s b o wd o w n to sim p le d e c o u p led lin e a r e q u a tio n s:
a b gx = ¡ g
v b wx v b w
c 20
;
a b gy = ¡ g
v b wy v b w
c 20
;
a b gz = ¡ g : (4 )
It is in d e e d rem a rk a b le th a t stro n g w in d s m a k e th e g o -in g sm o o th b y w a y o f d e co u p lin g th e m a ze o f c o u p le dn o n -lin e a r d i® e re n tia l e q u a tio n s in to a d o cile se t o f th reelin e a r eq u a tio n s! T o k e ep th in g s e v e n m o re sim p le , w e
a ssu m e th a t th e w in d is b lo w in g a lo n g th e d ire ctio n o fth e p itch in to th e b a tsm a n a lo n g th e n eg a tiv e X -a x is.F u rth er, w e a ssu m e th e b a ll is h it e ith e r h ig h o v er th e
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381RESONANCE April 2008
Fine leg
Longstop
Third man
Slipe
WK
Gully
Point
Cover
Squareleg
SqLU
Mid-wicketU
BowlerMid-off Mid-on
Long off Long on
On (Leg) sideOff side
S
NS
Y
X
2e
Box 1. Cricket: Fielding Positions for a Right-handed Batsman.
b o w le r o r p u lle d lo w o v er th e sq u a re (see B o x 1 ) le g u m -p ire S q L U . In th e ¯ rst c a se th e b a ll b a llo o n s u p o v e rth e b o w le r a n d th e w in d d ra g s it b a ck to w a rd s th e b a ts-
m a n . C u rio u s? L e t u s d e riv e a fo rm u la to d riv e h o m eth is id e a . T h e d o c ile se t o f th re e eq u a tio n s b e c o m es asw e et se t o f tw o :
a b gx = ¡ g
µv w g
x
c 0
¶2
;
a b gz = ¡ g :
This map shows the
batsman (S), straight
umpire (U), square leg
umpire (Sq L U), and
fielder positions. The tussle
between the bowler and the
batter is orchestrated by the
captain and his team of
players positioned stra-
tegically in the oval field.
The ellipse with major axis
2a has the batter and the
bowler at the foci separated
by 2e. The polar equation
of the ellipse using the
batter as the origin is r =
p/(1– cos), where p is
the boundary distance
along Y-axis and is the
ratio e/a. Tennis ball
cricket, however, bends
geometric rules to blend in
with the available space.
The sprawling city of Bangalore and suburbs can perhaps boast of the largest tennis ball cricket following in
the world with well over a million players. This revolution was fuelled in the 1960s with Test cricket legends
like B S Chandrashekar and G R Vishwanath participating actively in suburban tournaments. There is no doubt
that tennis ball cricket represents the most imaginative Indian innovation in sports and pastime in recent times.
(Adapted from http://en.wikipedia.org/wiki/fielder)
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382 RESONANCE April 2008
Figure 1. Flight paths in
vertical plane under strong
wind blowing left.
X
Z
O m ittin g su p e rsc rip ts a n d in teg ra tin g th e e q u a tio n s
v x = v x o ¡ ® g t;
v z = v z o ¡ g t;
w h e re ® = [v b w
x
c 0]2 ; a n d , v z o , v x o a re th e in itia l v e lo c ity
c o m p o n en ts. In teg ra tin g a g a in a n d elim in a tin g th e tim ev a ria b le t g iv es th e ° ig h t p a th . T h is w a s p ro v e d to b eg iv e n b y th e e q u a tio n
(x ¡ ® z )2
z v x 0 ¡ x v z 0
=2 (® v z 0 ¡ v x 0 )
g: (5 )
T h is e q u a tio n rep re se n ts a tilte d p a ra b o la . T h e a n g le o f
tilt is g iv e n b y a rc ta n (1 / ® ) w ith re sp ec t to th e g ro u n d .T h e w in d e ® e c t is lik e p la y in g o n a m o u n ta in slo p e.T h u s, stro n g w in d s a n d le v el p la y in g g ro u n d s d o n o tg o to g eth e r! S o m e ty p ica l tra jec to rie s a re sh o w n in F ig-u re 1 . O b serv e th e stra n g e b u t sp ec ia l c a se o f th e b a llm o v in g u p a n d d o w n a stra ig h t lin e . In th is ex tra o r-
d in a ry situ a tio n th e w in d re tu rn s th e b a ll b a ck to th eb a t. T h is is n o t a s o d d a s it se e m s w h e n th ere is n ow in d . A b a ll th ro w n v ertica lly u p co m e s d o w n to th esa m e p o in t e v e n tu a lly . U n d er lig h t b re ez e c o n d itio n s,th is sp e c ia l a n g le is a b o u t 8 0 d e g re es, a n d u n d er stro n g
w in d s a n g le s a s lo w a s 4 5 to 6 0 d eg ree s a re p o ssib le . U n -d e r m o d e ra te b ree z e, h o w e v e r th e b a ll e x ec u te s a b re e zylo o p re sem b lin g a n a irfo il sh a p e .
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383RESONANCE April 2008
Figure 2. Aerial view of
swinging ball under strong
wind blowing left.
X
Y
W e co n c lu d e d th e p re v io u s c la ssro o m a rtic le [1 ] w ith ase co n d e x a m p le o f a te n n is b a ll h it lo w o v e r th e sq u a rele g u m p ire . W e a re n o w in te re ste d in th e w a y th e b a ll
d rifts in th e d ire c tio n o f th e w in d w h e n v ie w e d fro mth e to p in th e X Y p la n e. T h ere fo re , w e n ee d o n ly tw oe q u a tio n s.
a b gx = ¡ g
µv w g
x
c 0
¶2
;
a b gy = 0 :
In te g ra tin g a fte r d ro p p in g su p erscrip ts, w e g e t
v x = ® g t;
v y = v y 0 = c o n st:
In th is ca se , th e w in d c a rrie s th e b a ll b eh in d th e p o si-tio n o f th e le g u m p ire b e fo re th e b a ll h its th e g ro u n d .L o o k in g fro m a b o v e , th e b a ll sw e ep s a p a ra b o lic a rc in
th e X Y -p la n e (F igu re 2 ), g iv e n a s fo llo w s
y 2 +
Ã2 v 2
y o
® g
!
x = 0 ; (6 )
3 . S p in n in g B a ll F lig h t : M o d e r a t e W in d s
In sp ire d b y th e le g e n d a ry fre e k ick ex e c u ted b y B e ck h a mo n 2 6 th J u n e 1 9 9 8 , C o o k a n d G o ® [2 ] e x p lo red th e c o m -b in a tio n o f p a ra m e ters fo r su c c essfu l so c ce r k ick s. T h eir
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384 RESONANCE April 2008
The magnus force is
directly proportional to
the cross product of
spin(s) and relative
velocity of the ball
with respect to air (R).
c a lcu la tio n s su g g est th a t B eck h a m k ick ed th e so c c er b a lla t a b o u t 1 0 0 k m / h w ith a sp in o f 6 0 0 rp m ! T h is k ickro se w ell o v e r a b e w ild e re d w a ll o f d e fen d ers e n ro u te
th e to p ed g e o f th e g o a l ( 6 0 ft y o n d e r, 8 ft u p a n d1 2 ft to th e le ft). T h e re is a tra d e o ® b etw e en sp ina n d tra n sla tio n w h ile b en d in g th e so c c er b a ll in ° ig h t.In ten n is b a ll c rick e t, h o w e v e r, sp in n in g sh o ts o c cu r b ych a n ce ra th e r th a n b y d esig n . T h e a d d itio n a l sp in in -
d u c ed a ero d y n a m ic fo rc e is th e fa m o u s m a g n u s e ® e ct.T h e m a g n u s fo rc e is d ire ctly p ro p o rtio n a l to th e c ro ssp ro d u c t o f sp in (s ) a n d re la tiv e v e lo c ity o f th e b a ll w ithre sp e c t to a ir (R ). T h e co n sta n t o f p ro p o rtio n a lity k s ,lik e th e d ra g c o e± c ie n t, d ep e n d s o n th e sh a p e , size a n dte x tu re o f th e b a ll. T h is a d d itio n a l a e ro d y n a m ic fo rce
g iv e s th e n e w eq u a tio n o f m o tio n ,
m a b g = (D + m g ) + k s s £ R : (7 )
D ra g fo rc e is p ro p o rtio n a l to th e sq u a re o f re la tiv e v e lo c -ity a n d a c ts o p p o site in d ire c tio n to th e re la tiv e v e lo c ityv e c to r. W e ta k e c w a s th e w in d sp e ed a n d w h e n th e w in dsp ee d a tta in s c o , th e d ra g b ec o m e s e q u a l to th e w eig h to f th e b a ll. T h u s, c o is a lso th e te rm in a l v e lo c ity o f th e
b a ll fa llin g fre e ly in th e v e rtica l d ire ctio n . W ith re sp ec tto th e sp in n in g b a ll, a sp in o f m a g n itu d e s o a n d w in dsp ee d c o p ro d u c e a m a g n u s fo rc e o f m g . B a sed o n C o o ka n d G o ® d a ta fo r so c c er b a ll[2 ], c o is a b o u t 2 5 m / s a n ds o a b o u t 6 0 0 rp m . It is ra th er a re m a rk a b le c o in cid e n ce
th a t c o fo r a so cc e r b a ll is a b o u t th e sa m e a s fo r a te n n isb a ll!
T h e b a tsm a n is a t th e o rig in o f c o o rd in a te s (X ,Y ,Z );X -a x is is a lo n g th e p itch a n d w in d is b lo w in g in th e
n e g a tiv e X d ire ctio n (i.e ., to w a rd s th e b a tsm a n ). W eta k e Y -a x is in th e v ertic a l d ire ctio n in th e se q u el; a n d ,th e refo re, th e Z -a x is is n o w p o in tin g a w a y fro m th e le gu m p ire . H o w e v e r, w e d o n o t re a lly n ee d a n y sp e c i cv a lu e fo r c 0 if w e u se n o n -d im e n sio n a l p a ra m e ters. T h e
re la tiv e v elo city o f th e b a ll w ith resp e ct to th e w in d is
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385RESONANCE April 2008
d e n o te d a s R = [(u + c w )2
+ v 2 + w 2 ]1 = 2 , w h e re u ; v ; w
re p resen t th e b a ll v elo city c o m p o n en ts. T h e g o v e rn in ge q u a tio n s o f m o tio n o f a sp in n in g b a ll o f m a ss m a re :
md u
d t= ¡ k d (u + c w )R + k s (s 2 w ¡ s 3 v );
md v
d t= ¡ k d v R ¡ m g + k s (s 3 (u + c w ) ¡ s 1 w );
md w
d t= ¡ k d w R + k s (s 1 v ¡ s 2 (u + c w )): (8 )
In th e a b o v e e q u a tio n s s 1 , s 2 , s 3 a re th e sp in co m p o n e n tsa b o u t X , Y a n d Z a x e s, re sp e c tiv e ly. It is c o n v e n ien tto n o rm a lize th e d ra g a n d m a g n u s fo rce w ith re sp ec tto th e w e ig h t o f th e b a ll b y n o rm a liz in g th e v e lo c ityc o m p o n en ts w ith resp e ct to c o a n d sp in c o m p o n en ts w ith
re sp e c t to s o . T h e n o n -d im e n sio n a l fo rm o f e q u a tio n s o fa sp in n in g b a ll b e c o m es
1
g
d u
d t= ¡ (u + c w )R = c o
2 + (s 2 w ¡ s 3 v )= c o s o ;
1
g
d v
d t= ¡ (v R + c o
2 )= c o2 + (s 3 (u + c w ) ¡ s 1 w )= c o s o ;
1
g
d w
d t= ¡ (w R )= c o
2 + (s 1 v ¡ s 2 (u + c w ))= c o s o : (9 )
T h e a b o v e se t re se m b le s V o lte rra { L o tk a eq u a tio n s u se dfo r a tm o sp h eric a n d e c o lo g ic a l m o d e llin g . T h ese e q u a -
tio n s a re o fte n e x p re ssed in th e K o lm o g o ro v fo rm :
d u
d t= u f (u ; v ; w );
d v
d t= v g (u ; v ; w );
d w
d t= w h (u ; v ; w ):
L o ren z d e v elo p ed sim ila r eq u a tio n s to o p e n th e d o o rs o fn o n lin ea r d y n a m ic s a n d ch a o s [4 ]. T h e g e n e ra l ca se o fth e sp in n in g b a ll ° ig h t re q u ire s 4 in itia l v e lo c ities (u o ; v o ;
w o ; c w ) a n d th re e sp in c o m p o n en ts (s 1 ; s 2 ; s 3 ). W e a s-
su m e in th is fo rm u la tio n th a t th e re is n o d ec a y in sp inw ith tim e (js j = c o n sta n t).
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386 RESONANCE April 2008
Figure 3. Flight paths for
increasingprojectionangle
(gentle wind blowing left).
0
20
40
60
80
100
120
80
-600
20
40
60
200 40 60 80 200 40 60 80 -40 -20 60400 20
40
0
20
60
80
120
100
140
100
120
Y
X
Y Y
X X
4 . S p in le s s F lig h t : M o d e r a t e W in d s
In th is sec tio n , w e re co n sid er th e ° ig h t w ith o u t sp in inth e v e rtic a l (X Y ) p la n e. O n c e a g a in , th e id e a is to e x -p lo re th e p o ssib ility o f th e b a ll retu rn in g to th e o rig info r a g iv en set o f v a lu es (u o ; v o ; c w ). U n lik e th e te n -n is b a ll ° ig h t u n d e r stro n g w in d , ° ig h t u n d e r m o d e ra tew in d o ® ers lim ited th e o re tica l sc o p e fo r d eriv in g c lo se d
fo rm so lu tio n s. It b e c o m e s n e cc e ssa ry to ex p lo re n u m e r-ic a l stra te g ie s to g a in in sig h t in to th e ° ig h t o f th e te n n isb a ll. F o r th e sa k e o f p ro v id in g a p h y sic a l fee l fo r th ep ro b le m , w e illu stra te th e rich v a rie ty o f ° ig h t p a th s b ya ssu m in g c w = 3 0 m / s , c o = 4 4 :3 m / s, R o = 1 0 0 m / s
in F igu re 3 .
T h e re is a sp ec ta c u la r d iv e rsity o f ° ig h t p a th s p o ssib led e p e n d in g o n th e in itia l a n g le o f p ro jec tio n (F igu re 3 ).
O b se rv e th e c a se o f a lo o p in g ° ig h t p a th . T h is lo o pg ra d u a lly e v o lv es in to a c u sp . A t a sp e c ia l a n g le o f p ro -jec tio n , th e te n n is b a ll re tu rn s to th e o rig in . A b o v e th issp ec ia l a n g le th e te n n is b a ll fa lls b eh in d th e b a tsm a n(a n d h o p efu lly n o t in to th e w ick etk ee p e rs h a n d s!).
T o ex p re ss re su lts u sin g n o n -d im en sio n a l q u a n titie s, w ec a n ta k e c o a s th e u n it o f sp e ed . W e d e ¯ n e n o n -d im e n -
sio n a l v e lo c ity r = (u 2 + v 2 )1 = 2
. T h e u n it o f le n g th is
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387RESONANCE April 2008
Figure 4. Return Paths [ ro=
0.226, o
= 64.2o, 43.5o,
10.75o ], [ ro= 2.26,
o= 56.3o,
38.7o, 9.69o ], [ ro
= 22.6, o=
52.03o, 37.2o, 9.64o].
w
0.30.1 0.2010.90.80.70.60.50.40.30.20 0.1
C =2.26
C =.68
C =1
w
w
w
0.3
C =2.26w
0.1
0
0.2
0.5
0.4
0.6
10.90.80.70.60.50.4
C =2.26w
C =1w
C =1
C =.68w
0.8
0.7
1
0.9
C =.68w
0
0.1
0.4
0.2
0.3
0.6
0.5
0.7
0.9
0.8
1
0.10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1X X
X
Y
Y Y
ta k en a s h o , th e h e ig h t a tta in e d b y th e b a ll w h e n th ro w nv e rtic a lly w ith a v elo city R o a ssu m in g a n e® ec tiv e v a lu eo f g ra v ity g ¤ = g (sin µ o + ® co s µ o ), w h e re µ o is th e a n g le
o f p ro jec tio n a n d ® = c w2 = c o
2 . W h en th ere is n o w in d
(c w = 0 ), it c a n b e sh o w n th a t h o = c o2
2 g ¤ ln (1 + R o2
c o2 ), fo r
v e rtic a l ° ig h t.
F igu re 4 p resen ts n o n -d im e n sio n a l tra jec to rie s fo r th reed i® e ren t n o n -d im e n sio n a l w in d sp e e d s c w = 0 :6 8 ; 1 :0 ; 2 :2 6 .It is in te re stin g to o b se rv e th e lo o p s g ettin g n a rro w e r
fo r d e c rea sin g v a lu e s o f r o . F o r v a n ish in g ly sm a ll r o ,th e sp e c ia l a n g le µ s = ¼ = 2 ¡ ® a s d isc u sse d in [1 ] fo rstro n g w in d s. T h e re tu rn p a th a lig n s w ith th e te rm in a lv e lo c ity v e c to r. T h e term in a l v e lo c ity v e c to r is v e rtic a lw h e n th e re is n o w in d . T h e c o n ce p t o f term in a l v e lo c -
ity in ste a d y h e a d w in d im p lies th a t th e b a ll is ca rrie d
CLASSROOM
388 RESONANCE April 2008
Figure 5. Spinning ball re-
turning to origin; uo= v
o= –
wo= .0226c
o; c
w= c
o= s
2.
b y th e w in d (u = ¡ c w ). T h e v e rtic a l c o m p o n en t o f th ete rm in a l v elo city is c o . T h ere fo re , th e te rm in a l a n g le µ t
sa tis¯ e s ta n µ t = (c o = c w ). T h e a n g les µ s , µ t a n d th e in itia l
a n g le o f p ro je c tio n µ o co m p le te th e k in em a tic p ic tu re o fth e ° ig h t.
5 . S p in n in g B a ll: S t r o n g W in d s
It is in tere stin g to stu d y th e ° ig h t o f sp in n in g b a ll u n d e rstro n g w in d b ec a u se th e e q u a tio n s sim p lify en o rm o u sly.U n d er stro n g w in d u ; v ; w < < c w ; a n d s 1 = s 3 = 0 ,
d u
d t= ¡ g
c w2
c o2
;
d v
d t= ¡ g ;
d w
d t= ¡ g
s 2 c w
c o s o: (1 0 )
T h is c a se is sim ila r to th e te n n is b a ll ° ig h t u n d e r stro n gw in d ex c ep t th a t th e p a th o f th e b a ll is n o t in th e X Y -p la n e (se e F igu re 5 fo r th e sp ec ia l c a se o f b a ll re tu rn in gto th e o rig in ). T h e m o tio n o f th e b a ll w o u ld b e e x ec u te d
o n a stra ig h t lin e in c lin ed eq u a lly to th e co o rd in a te a x es,if u o = v o = ¡ w o = :0 2 2 6 c o ; c w = c o = s 2 . T h e re su lt issh o w n in th e to p a n d sid e v ie w s in F igu re 5 .
X
0.02-0.02 00
0.02
0.060.04 -0.02
0.06
0.04
0.040
0 0.02
0.02
0.06
0.06
0.04
X
Z Y
CLASSROOM
389RESONANCE April 2008
Figure 6. Spinning ball re-
turning to the origin: Mod-
erate winds.
Y
10
0
0
5 10
5
15
25
20
2015 25 3530 40
35
30
30
10
0
0 10 20
20
30 40
40
X X
Z
6 . S p in n in g B a ll: M o d e r a t e W in d s
W h e n th e b a ll a n d w in d sp e ed s a re o f th e sa m e o rd er,th e tra je cto ry o f th e b a ll is sh o w n in F igu re 6 . T h e b a lle x e c u tes a 3 -D lo o p b e fo re re tu rn in g to th e o rig in fo rth e c o n d itio n s u o = v o = ¡ w o = c o , c w = c o . T h is° ig h t tra je c to ry h a s c o n sta n tly ch a n g in g cu rv a tu re a n dto rsio n .
7 . S p in n in g B a ll: A e r ia l V ie w
W in d a n d g ra v ity c o m p lic a te m a tte rs. T h e re a re situ -a tio n s w h ere g ra v ity a n d w in d b e co m e re la tiv e ly u n im -p o rta n t. T h e a b se n ce o f g ra v ity a n d w in d , a s w e w ill se e,b rin g s o u t a rich d iv e rsity o f ° ig h t p a th s a s w ith frisb e e sa n d b o o m e ra n g s. L e t u s im a g in e th a t th e b a ll is in a
g ra v ity - a n d w in d -fre e en v iro n m en t. L e t u s th ro w th eb a ll sp in n in g a b o u t th e Y -a x is in th e X Z -p la n e a n d a lo n gth e X -a x is w ith v = w = 0 . L e t s 2 6= 0 ; s 3 = s 1 = 0 .T h e eq u a tio n s g o v e rn in g th e m o tio n o f th e b a ll red u ceto :
d u
d t= ¡
u R
c o2
g + gs 2 w
c o s o;
d w
d t= ¡
w R
c o2
g ¡ gs 2 u
c o s o
: (1 1 )
CLASSROOM
390 RESONANCE April 2008
The absence of
gravity and wind as
we will see, brings
out a rich diversity
of flight paths as
with frisbees and
boomerangs.
P u ttin g u = R c o s µ a n d w = R sin µ , w e c a n sh o w th a t
d R
d t= ¡ g
R 2
c o2
:
U p o n in te g ra tio n w e g et
R (t) = R oc o
2
c o2 + R o g t
:
T h e co rresp o n d in g so lu tio n fo r µ (t) = ¡ g s 2
c o s ot is a s fo l-
lo w s
u = R o
c o2
(c o2 + R o g t)
c o s(¡g s 2
c o s o
t);
w = R oc o
2
(c o2 + R o g t)
sin (¡g s 2
c o s ot): (1 2 )
T a k in g a = (c o s 2 )= (R o s o ) a n d c = (g s 2 )= (c o s o ); th e ° ig h tp a th in th e X Z -p la n e is g iv e n in term s o f sin e a n d c o sin ein te g ra ls.
x (t) =c o
2
g((S i(a + c t) ¡ S i(a ))sin (a )
+ (C i(a + c t) ¡ C i(a ))co s(a ));
z (t) =c o
2
g((C i(a + c t) ¡ C i(a ))sin (a )
¡ (S i(a + c t) ¡ S i(a ))c o s(a )): (1 3 )
In th e a b o v e so lu tio n S i(x ) a n d C i(x ) a re th e d e ¯ n ite
in te g ra ls b e tw ee n th e lim its 0 to x o f th e fu n c tio n s sin (x )x
a n d c o s(x )x , resp e ctiv e ly. W e n o tic e th e g ra d u a l d e ca y
in sp e ed w ith tim e a s th e b a ll sp ira ls to w a rd s its ¯ n a lp o sitio n . T h is ¯ n a l p o sitio n d ep e n d s o n th e in itia l v e -
lo c ity a n d sp in . T h e sp ira l v a ria tio n o f th e v elo city isp o rtra y e d in F igu re 7 . N o te th a t th e v e lo c ity o f th esp in n in g b a ll v a ries in v ersely w ith tim e e la p se d v a n ish -in g ev e n tu a lly. T h e sp in n in g b a ll a p p ro a ch e s its ¯ n a ld e stin a tio n w ith c o o rd in a te s (6 8 .7 ,{ 1 2 4 .3 )m w h en th e
CLASSROOM
391RESONANCE April 2008
Figure 7. Spinning ball spi-
ral in XZ-plane and veloc-
ity plot for Ro= c
oand s
2=
so.
120
ZD
ista
nce
(m)
-80
-140
-180-20 0
-160
-120
-100
X Distance (m)
6020 40 80 100
-60
-40
-20
0SPIROD IN ABSENCE OF WIND
VV
eloc
ity
alon
gY
-0.1
U Velocity along X
-0.4
-0.5140 -0.2-0.4 0
-0.3
-0.2
0.60.2 0.4 0.8 1.0
PONCARE PLOT IN ABSENCE OF WIND
0
0.1
0.2
There is a dramatic
transformation in the
flight path in the
presence of a slight
tail wind behind the
spinning ball.
b a ll is th ro w n a t sp e e d u o = c o w ith sp in s 2 = s o . T h e se¯ g u re s a re sim ila r to sa te llite p ic tu re s o f cy c lo n es.
8 . T a il/ h e a d W in d E ® e c t : A e r ia l V ie w
T h e re is a d ra m a tic tra n sfo rm a tio n in th e ° ig h t p a th inth e p resen c e o f a slig h t ta il w in d b e h in d th e sp in n in g
b a ll. T h e v e lo c ity d ia g ra m is sh ifted b y c w , b u t o th e r-w ise a p p e a rs sim ila r to F igu re 7 . T h e re su ltin g ° ig h tp a th F igu re 8 , h o w ev er, is d ra stic a lly d i® ere n t. T h e ta ilw in d ta k es th e b a ll a lo n g a d isto rte d c y clo id . T h is p a th -c a lle d sp iro id , b len d in g sp ira l w ith th e c y clo id , d isp la y s
e x o tic fe a tu re s u n im a g in a b le w ith o u t sp in . D e p e n d in go n th e m a g n itu d e o f th e ta il w in d , th e ° ig h t p a th c a ne x h ib it b o th co n tra cte d a n d e x te n d e d ch a ra cte ristic s o fa c y clo id . T h is is c le a rly e v id e n t th ro u g h th e lo o p s a n dc u sp s in F igu re 8 . O b se rv e th e ¯ n a l o ® se t o f th e b a ll
fro m th e o rig in a l d ire c tio n o f ° ig h t (w h ich d e p e n d s o n lyw ea k ly o n w in d sp e ed ). T h ese re su lts a re a t ¯ rst c o u n te rin tu itiv e, b u t it sh o u ld b e n o te d th a t w e h a v e u se d ala rg e v a lu e o f s 2 = s o to d em o n stra te th e se e x o tic v a rie tyo f sp in n in g b a ll tra jec to rie s. T w o d i® e re n t m a g n itu d e s
fo r th e ta il w in d a n d o n e v a lu e fo r h e a d w in d illu stra te
CLASSROOM
392 RESONANCE April 2008
-40
0
2500
-20
ZD
ista
nce
(m)
-120
-160
-180-1000
-140
-80
-100
-40
-60
X Distance (m)-800 -600 -200-400
0
-140
-120
0
ZD
ista
nce
(m)
-80
-100
-60
SPIROD UNDER HEAD WIND
X Distance (m)500 1000 20001500
200
40003000
-160
-180200100 300
ZD
ista
nce
(m)
-100
-120
-140
-60
-80
800
X Distance (m)
600500 700 1000900
-20
0SPIROD UNDER TAIL WIND SPIROD UNDER TAIL WIND
-20
-40
0
Figure 8. Aerial view of spinning ball
path: tail winds (co/10 and c
o/3) (top)
and head wind (co/10)(bottom). Note:
Ro= c
oand s
2= s
o.
h o w th e sp in n in g b a ll sn a k e s a lo n g u n til th e ta il/ h e a dw in d s d o m in a te th e d y n a m ic s.
In th e co n te x t o f te n n is b a ll c rick e t o r so c c er, a p la y e ra tte m p tin g to c a tch a sp in n in g a n d sn a k in g b a ll w illh a v e to b e a g ile a n d a le rt. T h is e ® e c t a lso m a k e s itp o ssib le fo r p itch e rs a n d b o w le rs to e x p lo it w in d a n dsp in to c o n fu se th e b a tter. S o , in ten n is b a ll c rick e t, in
a d d itio n to th e le g -sp in a n d o ® -sp in , w e h a v e th e sn a k in gs-sp in ! T h e se sp iro id a l p a th s a re a lso o b se rv e d d u rin gsw irlin g d u st sto rm s a n d c y c lo n e s.
9 . S u m m a r y
T h o u g h n o t ju st lim ited to so c c er a n d c rick e t, sp in is
CLASSROOM
393RESONANCE April 2008
Suggested Reading
[1] K R Y Simha, Tennis ball
flight under strong wind,
Resonance, Vol.7, No.11,
November 2002.
[2] B G Cook and J E Goff,
Parameter space for suc-
cessful soccer kicks, Eu-
ropean Journal of Phys-
ics, Vol. 27, pp.865–874,
2006.
[3] R D Lorenz, Spinning
Flight: Dynamics of
Frisbees, Boomerangs,
Samaras, and Skipping
Stones, Springer 2006.
[4] S H Strogatz, Nonlinear
Dynamics and Chaos,
Westview Press, p.301,
1995.
u se d b y p la y e rs to g a in a n a d v a n ta g e o v er th e ir o p p o -n e n ts. T h is m a k e s th e u n d ersta n d in g o f th e d y n a m ic so f th e sp in n in g b a ll ev e n m o re im p o rta n t a s it c o u ld
h e lp p la y ers im p ro v e th e ir g a m e . T h e e® ec ts o f sp in a rec le a rly se en in g a m e s lik e ten n is a n d ta b le te n n is (T T ).H e re th e p la y e r c a n im p a rt 3 k in d s o f sp in s to th e b a ll.
F o rw a rd sp in o r to p sp in : T T p la y e rs w ill k n o w h o w
d i± c u lt it is to k e ep th e b a ll o n th e sm a ll ta b le w h e nh ittin g a fa st sh o t. T h is p ro b le m is o v e rc o m e b y im -p a rtin g a to p sp in to b a ll w h ile h ittin g a fa st sh o t. T h eto p sp in c a u ses th e b a ll to d iv e , h e n ce su rp risin g th eo p p o n e n t a n d p itch in g b efo re h e e x p e c ts it.
B a ck w a rd sp in : T h e b a ck w a rd sp in im p a rte d b y a p la y e rc o u n ters g ra v ity (th e m a g n u s e ® e ct) a n d th is c a u ses th eb a ll to ° o a t in th e a ir lo n g er th a n ex p e c ted . In th is
w a y ev en a slo w sh o t c a n b e m a d e to cro ss th e n e t b yim p a rtin g a b a ck w a rd sp in to th e b a ll.
S id e sp in : S id e sp in is b est o b se rv e d d u rin g a se rv e inT T . W h en th e p la y er im p a rts a sid e sp in to th e b a ll th e
b a ll sw in g s sid e w a y s (m a g n u s e ® e ct) a n d m o v e s fu rth e ra w a y fro m th e o p p o n e n t, m a k in g it to u g h e r fo r h im tore ce iv e th e se rv e .
T h u s, a e ro d y n a m ic v iew o f sp o rts fu rn ish es a d ee p e ra p p re c ia tio n o f th e p la y e rs sk ills a n d p e rse v e ra n c e tom a ste r w in d , sp in a n d g ra v ity . S o fa r w e d iscu ssed th ed y n a m ic s o f sm o o th sp in n in g b a lls. B u t so m e b a lls (e .g .,g o lf b a lls) h a v e d im p le s o n th em , so m e a re n o t c irc u la r
(ru g b y b a lls); th e d y n a m ics o f th e se b a lls a re v e ry d i® e r-e n t a n d m u ch m o re ch a lle n g in g ! T h e in itia l in sp ira tio nfo r th is a rtic le ca m e fro m o b se rv in g th e sa m a ra se e d ssp in n in g w h ile fa llin g d o w n fro m th e tre e. P re se n tly,th e re is a te a m o f stu d e n ts d e sig n in g a pa ra co p ter b a se d
o n th e sp in n in g se e d p rin c ip le to a ch iev e a te rm in a l v e -lo c ity o f le ss th a n a b o u t 4 m / s fo r a 1 0 0 k g p a ck a g ed ro p p e d fro m a h e ig h t o f 1 0 0 m .
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