american journal physics

8/17/2019 american journal physics

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A classical experiment revisited: The bounce of balls and superballsin three dimensions

Antonio Doménecha)

Department of Analytical Chemistry, University of Valencia, Dr. Moliner, 50, 46100 Burjassot (València),Spain

Received 14 January 2004; accepted 22 July 2004

A description of the inelastic collision of a ball when it bounces on a rigid horizontal surface with

arbitrary initial spin conditions is presented. I consider cases where rebound occurs with and without

sliding, and when the ball grips the surface. Two rebound models are discussed in which friction isdescribed in terms of the static and dynamic coefficients of friction and the friction is expressed in

terms of horizontal coefficients of restitution. The azimuthal and vertical deviation angles of the ball

after impact are predicted as a function of the incident angle. We present data from an experiment

in which a ball is launched horizontally from the edge of the laboratory bench and then rebounds on

a horizontal surface. Ordinary balls exhibit two rebound regimes, with and without sliding, and can

be satisfactorily described by the first model. Superballs exhibit a grip behavior whose description

requires the use of the second model. © 2005 American Association of Physics Teachers.DOI: 10.1119/1.1794755

I. INTRODUCTION

A common goal in ball sports is to get a ball to bounce atan oblique angle on a rigid surface. This goal is of interest ingolf and tennis, but also in basketball, soccer, and handball.The study of collisions also is relevant in the study of non-linear dynamical systems1 and granular matter.2,3

Approximate solutions of the dynamics of a ball bouncingon a floor have recently been described. Brancazio4 analyzedthe bounce of a basketball having an initial forward/ backward spin by assuming that the collision is perfectlyelastic and that the ball does not skid just after the impact.Brody5 studied the bounce of a tennis ball and assumed thatthe collision is inelastic in the vertical direction and com-pletely inelastic in the horizontal direction. Garwin6 de-

scribed the bounce of a superball by assuming that the col-lision is perfectly elastic in both the vertical and horizontaldirections. More recently, Cross7– 9 described the oblique im-pact of a ball of mass on a block and considered the impactto be inelastic. Following Ref. 9, the consideration of frictionforces leads to three possible regimes of motion: pure slid-ing, slide then roll, and slide then grip. The grip or slip-gripbehavior occurs in some instances, namely, in the rebound of tennis balls and superballs under certain conditions. Thegrip-slip behavior is characterized by the appearance of alarge spin after an impact.6,8,9

Alternatively, Maw et al.10 described the oblique bounceof a solid elastic sphere in terms of a numerical model. Inthis approach, the contact circle is divided into small annuli,

some of which grip the surface and some of which slip lead-ing to results similar to those of Cross.9 The models of Refs.9 and 10 provide theoretical predictions in close agreementwith experimental data for the oblique bounce of tennis balls,golf balls, baseballs, and basketballs without initial spin onsmooth and rough surfaces. In this context, Metha and Luck studied the evolution of a bouncing ball on a vibrating plat-form and expressed the inelasticity of the impact in terms of the classical coefficient of restitution.1

In this paper, a ball with an initial arbitrary spin thatbounces obliquely on an infinitely massive surface is studied.It is well known that a ball launched forward with an initial

forward spin gains speed after the bounce, whereas a ball

initially launched with backspin loses considerable speed af-ter the bounce and may even reverse its direction of motion,that is, it may bounce backward.4 If an additional spin alonga horizontal axis parallel to the horizontal velocity of the ballis imparted to the ball, it deviates from the initial direction of motion. As a result, a variety of rebounds can be obtained.

These effects must be attributed to friction forces actingduring the impact, and were first treated by Whittaker in190411 and described in detail by Keller.12 Classical ap-proaches to the inelastic impact with friction of rigid bodieshave been developed by Brach13 and Kane and Levinson.14

In these formulations,12–14 the inelasticity of the impact isexpressed in terms of the classical coefficient of restitution,e, which is the negative of the ratio of the relative normal

velocity after impact to that before impact.15

These authors describe two possible regimes of impactderived from the action of friction forces with and withoutsliding between the contacting surfaces. When sliding existsbetween the colliding bodies, the friction forces are ex-pressed in Ref. 14 in terms of the classical static and dy-namic coefficients of friction and the coefficient of restitu-tion; Ref. 13 uses an equivalent coefficient of friction withsimilar meaning and adds a moment coefficient of restitutionto the classical coefficient of restitution. Following Brach,13

the moment coefficient of restitution can be defined as theratio of the angular velocities of the colliding bodies afterand before the impact. In both cases it is assumed that forsufficiently large friction forces, a nonsliding regime is at-

tained; in this regime the relative tangential motion of thecontacting points ceases during the impact.

Following these approaches, the impulsive forces actingthroughout the oblique rebound of a ball can be described asa normal retarding force, F n , plus a tangential friction force,

F f . For large incident angles, relative tangential motion al-ways must exist during the time of contact between the balland the horizontal surface; after the contact the ball slidesthroughout the bounce.

The approach presented here compares two alternative for-mulations of the impact between rigid bodies to describe thegeneral case in which a ball having an arbitrary spin is

28 28Am. J. Phys. 73 1, January 2005 http://aapt.org/ajp © 2005 American Association of Physics Teachers


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launched against an infinitely massive horizontal surface.The problem can be formulated in terms of classical frictionforces. If we use the classical formulation first developed byAmontons and further developed by Coulomb,16 frictionforces are treated as proportional to the normal retardingforce. In this case a collision in the sliding regime is obtainedand the ball rebounds with a sliding plus rolling motion.Alternatively, a generalized non-Amontons–Coulomb re-gime is described by coefficients of restitution in three di-mensions, e x , e y , e z . This description is consistent with theapproaches of Ref. 13 in which a moment coefficient of res-titution is defined, Refs. 7–9 in which horizontal and verticalcoefficients of restitution are introduced,7– 9 and Ref. 3 whichdescribes the impact of small, nearly identical spherical par-ticles in terms of the Newtonian coefficient of normal resti-tution and coefficients representative of the frictional prop-erties of contact surfaces with and without negligible sliding.

As in these approaches,7–14 the coefficients of restitutionand friction will be assumed to be constant. This assumptionis a simplification; the coefficients of restitution and frictiondepend on several factors, including the elastic properties of the materials, surface state, relative velocity, and the balldiameter.15,17 The validity of this assumption will be dis-cussed in relation to experimental data.

We follow the methodology previously developed for theimpact of disks18 and billiard balls,19 and predict the verticaland azimuthal angles of rebound from the incident angle forarbitrary angular rotation rates in each one of the regimes,sliding, nonsliding, and grip-slip.

The bounce of a uniform sphere launched after rollingwithout slipping along a horizontal plane is typically used asan example of inelastic collisions. In a classic experiment, arolling ball is projected horizontally from the edge of thelaboratory bench and then rebounds from a horizontal sur-face. The horizontal distances xo and x covered prior to the

first and second bounces satisfy the relation x2ex o , where

e is the coefficient of restitution for the impact between theball and the floor. A generalized version of this experiment

that can be used in classroom discussions and intermediatelab experiments is described. Different initial spins are im-parted to the incident ball by the collision with an auxiliarycue ball. The angles of impact and rebound and the distances x and x o can easily be measured to test the theory.

II. THEORY

A. General equations of motion

Let us consider a homogeneous sphere of mass m and

radius R launched obliquely with initial center-of-mass linear

velocity vo against an infinitely massive horizontal surface.

As can be seen from Fig. 1, if the sphere is launched alongthe x axis, the velocity components of the center of mass of

the sphere are voxvo sin , voy0, and vozvo cos . Af-

ter the impact, the sphere travels with a velocity v in a di-

rection that may be separated from the xz plane character-ized by a vertical angle and an azimuthal angle, . Therebound velocity components, v xv sin cos , v yv sin sin , v zv cos , can be derived from the time in-tegrals of the force impulse over the time of impact:

F ndt mv cos vo cos , 1

F f xdt mv sin cos vo sin , 2

F f ydt mv sin sin . 3The normal component of the velocity of the point of con-

tact of the sphere after the bounce with the horizontal planewill be e times its value prior to the impact, that is, v cos

evo cos . Hence, the integrated normal retarding forcebecomes:

F ndt mvo1ecos . 4If the incident sphere has initial angular velocities ox , oy ,

oz along the x , y , z axes, the direction of the horizontal

components of the friction force, F f x , F f y , will depend onthe direction of the horizontal components of the velocity atthe point of contact just before the bounce and are givenby voxvox R oy , voyvoy R ox . If we assume, as hasbeen previously discussed,19 that the direction of the frictionforce is the same as that of the horizontal velocity at thepoint of contact at the beginning of the bounce see Fig. 1,we can write F f xF f cos , F f yF f sin , where,

tan F f y

F f x

R ox

vox R oy. 5

The direction of the friction force depends on the initial spinas described in Ref. 4. In particular, if the ball is thrown withpure rolling motion, R ox0, R oyvo , there is no fric-tion force at the point of contact.

In general, a small deformation occurs at the contactingregion. It is assumed that the normal force acts verticallythrough a line passing a distance r behind the center of mass

of the sphere.9 Then, an additional torque appears which willmodify the angular velocity of the ball. For simplicity, it willbe assumed that r R; that is, the deformation effects will beneglected. Then, conservation of angular momentumyields,20

I x xo mv yvoy , 6

I y oy mv xvox , 7

where I (2/5)mR 2 is the moment of inertia about an axis

through the center of the sphere, and x , y , z , are thecomponents of the angular velocity of the ball just after the

Fig. 1. Schematic of the bounce of a ball on an infinitely massive horizontal

surface for arbitrary initial spin. V ox , V oy represent the horizontal compo-

nents of the velocity of the ball just before the rebound event.

29 29Am. J. Phys., Vol. 73, No. 1, January 2005 Antonio Doménech


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bounce. This treatment can easily be extended to balls usedin sports by replacing the moment of inertia of a uniformsolid sphere by that for a thin spherical shell, I

(2/3)mR 2.If we combine Eqs. 1–7, we obtain a description of the

motion of the ball after the bounce provided that the relationbetween F f and F n is known. We will consider two descrip-tions of the rebound, namely, a Amontons–Coulomb-typefriction force and a normal coefficient of restitution the -e

model, and non-Amontons–Coulomb friction force the e -e

model. The latter model introduces coefficients of restitutionin three dimensions, e x , e y , e z , the former being the sameas the usual coefficient of restitution in the normal direction,e . As a result, the friction force becomes proportional to thehorizontal velocity of the incident ball before the rebound.The case of impact without sliding can be derived from both -e and e -e models.

In the following, equations for the linear and angular ve-locities and rebound and azimuthal angles after the impactwill be given for the case in which ox0. Note that in allcases, the time integral of the normal force equalsmvo(1e)cos , where v zevo cos , and z oz .

B. Rebound with Amontons –Coulomb friction

If we assume that friction can be described by aAmontons–Coulomb force when sliding exists at the re-bound, the friction force can be taken to be times thenormal force, that is, F f F n . We combine this relationwith Eqs. 1–7 and obtain:

v xvosin 1e cos cos , 8

v yvo 1e cos sin , 9

R x R ox5

2 v o 1ecos sin , 10

R y R oy52 v o 1ecos cos . 11

The azimuthal angle is given by,

tan 1esin

tan 1ecos , 12

and the angle of rebound becomes

tan tan 1e cos

e cos . 13

Equations 8–13 are notably simplified for the most com-mon case in which ox0, where tan 0, and,

tan 1/ etan 1ecos / e . 14

Note that there are two possible situations. For a large for-ward spin, vox R oy0, cos 1, and the rebound angleis larger than the impact angle. For backward or small for-ward spins, vox R oy0, cos 1, and the friction force isopposite to the translational motion. The rebound angleequals the impact angle when tan (1e)/(1e). As a

special case, when vox R oy1, that is, if the incident ballis launched horizontally with pure rolling motion, no frictionforce arises, and then 0 and tan (1/ e)tan . This result

comes from Eq. 13 by taking 0.

When nonsliding takes place at the rebound, v x R y0, v y R x0, and the linear and angular velocities after

the impact can be obtained from Eqs. 1–7. If R ox0,the rebound angle satisfies

tan 5/7e2/7e R oy / vo sin tan . 15

The transition from the sliding regime of rebound to non-sliding occurs when the friction force F f oF n , where ois the static coefficient of friction. The transition occurs at alimiting impact angle, L , given by

tan L7/2 o1e cos / 1 R oy / vo sin .16

C. Rebound without Amontons –Coulomb friction

Let us consider friction forces that are not described by theAmontons–Coulomb law. To obtain a description of suchforces, the horizontal coefficients of restitution, e x , e y , canbe defined in a way similar to the normal coefficient of res-titution, ee z .

15 That is, the horizontal restitution coeffi-cients will be defined as the proportionality constants be-tween the relative horizontal components of the velocity of the point of contact before and after the impact:

e xv x R y

vox R oy, 17

e yv y R x

R ox. 18

This formulation replaces the Amontons–Coulomb force bya friction force defined from these coefficients of restitution.From Eqs. 1–7, 17, and 18, we obtain:

v x 572

7 e x vo sin 27 1e x R oy , 19

v y

2

7 1e y R ox , 20

R y5

7 1e xvo sin 27

5

7 e x R oy , 21

R x 275

7 e y R ox . 22

From Eqs. 19–22 we obtain the corresponding generalexpressions for the horizontal and vertical angles of rebound:

tan 2/71e y R ox

2/71e x R oy5/72/7e xvo sin , 23

tan

5/72/7e xtan 2/71e x R oy / vo cos

e z cos .

24

We note that, in general, sliding exists between the points of contact of the ball and the floor. If nonsliding occurs seeEqs. 17 and 18, e x0 and e y0, and for R xo0, Eq.24 reduces to Eq. 15.

Different rebound regimes can be defined depending onthe values of the coefficients of restitution. These regimes



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will be discussed for the case in which R ox0 and R oy0. Then, 0, that is, there is no horizontal deviation of the ball after the rebound. We distinguish:

a The ordinary regime, e x0, where the absolute value

of v

x is larger than R y ; the ball slips along thebounce and a relatively low spin is acquired during theimpact.

b The nonsliding regime, e x0. In this case the ballabandons the horizontal surface with pure rolling mo-tion, R yv x . The angle of rebound is tan (5/7e)tan , in agreement with Eq. 15. The horizon-tal velocity and the angular velocity become, respec-tively, v x(5/7)vo sin , R y(5/7)vo sin .

c The ‘‘grip’’ or ‘‘superball’’ regime, e x0 in which alarge spin is imparted to the ball throughout the impact: R yv x .

The ideal elastic frictionless case is given by e xe y1, e z1. Accordingly, tan tan , that is, the angle of rebound equals the angle of impact. This situation is equiva-lent to that described in the -e model by considering e

1 and 0.

We emphasize that the -e and e -e models differ signifi-cantly in the expression for the friction forces. For the case inwhich the ball is initially launched without spin, the -e

model predicts an integrated force of mvo (1e z)cos ,

while the e-e model predicts a value of mvo(2/7)(1

e x)sin . Given the implicit assumption that the frictionand restitution coefficients are constant, it appears that thereis no possibility for reconciling the two formulations.

D. The horizontal launch experiment

To test the different models a classic experiment was done.As depicted in Fig. 2, it consists of the horizontal launch of a ball from a horizontal track. After the ball leaves the track,the ball travels through the air and strikes the floor. To impartdifferent spins to the ball, an auxiliary cue ball was used. Asdescribed in Sec. III, the impact and rebound angles caneasily be measured as well as the horizontal distances cov-ered by the ball before ( x o) and after ( x) its rebound. If the

initial velocity of the ball, u ovo sin , remains constant,experiments with variable height can test the proposed mod-

els from angle measurements and from the measurement of x

and x o . For rebounds with sliding, the -e model yields

x2e xo 1 1eguo

2 xo , 25where the sign corresponds to rebounds with vox R oy0, and the sign to rebounds with vox R oy0. Equa-

tion 25 predicts that the ratio x / xo provides a linear func-

tion of x o and is equal to 2e at x o0.

The e-e model leads to the general relation:

x2exo 572

7 e x

2

7 1e x R oy . 26

Equation 26 predicts that a plot of x / x o as a function of x ois a straight line with zero slope and a value at the origin thatdepends on e x and oy .

For rebounds without sliding, both the -e and e -e mod-els lead to

x2exo572

7 R oy

u o . 27

Here, the ratio x / xo is independent on x o and depends on the

coefficient of restitution and the initial angular velocity oy .As expected, both the -e and e -e models predict if a ballis launched with initial pure rolling motion, that is, R oy / u o1, x / x o2e , in agreement with the well-knownexperiment of elementary mechanics. Hence, the post-rebound linear and angular velocities of a ball launched witha particular value of R oy can be calculated from the actual

x and xo values and those measured when that ball is

launched with R oy1. We denote these values as x* and

xo* and write

v xu o x / xo xo* / x*, 28

v yg x o /2u o x* / xo*, 29

R y R oy5/21 x / xo x o* / x* , 30

where the integrated friction force can be calculated from:

F f xdt mu o1 x / x o xo* / x*. 31Equations 28–30 are independent of the nature of thefriction force and, consequently, can be used for testing the -e and e-e models from measurements of the linear andangular velocities. Additionally, such velocity measurementscan be used to determine the dependence of the coefficientsof restitution and friction on different parameters such as thevelocity and the diameter.

III. EXPERIMENTAL ARRANGEMENT

The experimental arrangement is depicted schematicallyin Fig. 2. A manufactured aluminum track width 1 cm wasplaced on a lab bench, with one end of the track flush withthe edge of the bench. The balls were first allowed to rollfrom rest along the track with the help of an auxiliary slantedrail. After rolling off the track, they traveled through the air,finally striking the floor, thus obtaining bounces with initialforward spin.

To obtain rebounds with initial backspin, a ball, initially atrest, was struck head-on by an auxiliary cue ball rolling from

Fig. 2. Schematic representation of the experimental arrangement. The ball

travels a distance x o while dropping a vertical distance h , and then rebounds

traveling a distance x along the horizontal surface.



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the inclined track. The ball was placed at a known distance d from the end of the track, thereby obtaining different valuesof oy with ox0. To study rebounds without initial spin,the ball was launched from the edge of the track with thehelp of a spring-operated shooter. To obtain the condition R oy / vox1, the track was replaced by a steel tube of di-ameter 5 cm.

Data were obtained by varying the height of the bench butensuring that a constant velocity at the bench edge occurs.The initial velocity of the ball was adjusted in each series of

experiments to 1.00

0.05 m/s. In all the experiments, theincident and rebound angles were measured as well as thehorizontal distances covered by the ball before its rebound.

The trajectories of the balls were determined from photo-graphs recorded by a camera placed either in a vertical andhorizontal position following the method described in Refs.18 and 19. Steel ball bearings of diameters 1.22, 1.28, 1.51,1.71, 2.00, 2.50, and 2.85 cm, a golf ball mass 45.6 g, di-ameter 4.26 cm and a superball mass 46.4 g, diameter 4.60cm were used. To test the cases in which small and largesurface deformation occurs, experiments were performed ona ceramic floor, a hard wood panel, and a plastic-type floor.The effect of air resistance could be neglected because of theball’s small size, high density, and short flight time.21

IV. RESULTS AND DISCUSSION PRIVATE

A. The sliding-nonsliding transition

The existence of a transition from a rebound without slid-ing to a rebound with sliding was tested. In Fig. 3 the ex-perimental values of the rebound angles are shown for a asteel bearing, b a golf ball, and c a superball bouncing ona polished ceramic floor without initial spin ( R ox0,

R oy0) with different incident angles but the same hori-zontal velocity at impact. If sliding exists in these circum-stances, the -e model predicts from Eq. 14 that plots of

tan versus tan will yield straight lines of slope 1/ e and

ordinate at (1

e)/ e. In contrast, the e-e model predictsfrom Eq. 24 that plots of tan versus tan will yield

straight lines passing through the origin with slope 5/7e z(2/7)(e x / e z). The nonsliding regime, common to both

models, corresponds to linear tan versus tan plots passing

through the origin with slope 5/7e see Eq. 15.For the steel bearing Fig. 3a, the data obtained at low

impact angles fit well with Eq. 15 by using e0.78. For

incident angles larger than approximately 24°, the data agreewell with the theoretical sliding graph obtained by lettinge0.78 and 0.07 to Eq. 14. The agreement between the

e values calculated for the two regions suggests that the -emodel applies, and the sliding regime occurring at high im-

pact angles.If we assume that the value of the static coefficient of friction 0 is close to that of the dynamic coefficient , thetheoretical critical incident angle at which the transition fromone rebound regime to another is close to 24°. The data do

not allow for the observation of a discontinuity in the tan

versus tan plot, required by the condition represented by

Eq. 16, thus suggesting that the value of 0 must be closeto that of .

A similar result was observed for the golf ball Fig. 3b.Here, the data agree with the theoretical prediction withe0.90, 0.12. With these parameter values, the transi-

tion angle is 39°. Again, the response at low impact anglesis close to that predicted by the nonsliding condition for bothmodels, that is, by taking e

x0 with e

z0.90.

However, for the superball, only one well-defined regionwas observed in the tan versus tan plots see Fig. 3c,corresponding to a straight line passing through the origin.By applying the nonsliding condition, we obtain an unrealis-tic e z value of 1.17, suggesting that this regime does not

occur under our experimental conditions. If we use e z0.96, close to that reported for the rebound of superballsagainst hard surfaces,6,8,9 our experimental data are consis-tent with Eq. 24 if we use e x0.50. These values are rea-

sonable, suggesting that the e -e model holds for the reboundof superballs.

Fig. 3. Experimental data filled circles and theoretical predictions for the

rebound of a a steel bearing, b a golf ball, and c a superball on a

ceramic floor without initial spin. Continuous lines: theoretical plots from

Eq. 15 using a e0.78 and b e0.90. The dotted lines correspond to

Eq. 15 with a e0.78, 0.07 and b e0.90, 0.12. The continu-

ous line in c corresponds to Eq. 24 with e x0.50.



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B. Influence of the initial spin

Experimental data concerning the horizontal distancescovered by the ball before and after the rebound provide a

sensitive method for determining not only the transition fromsliding to nonsliding rebounds but also the influence of theinitial spin. This experiment is of pedagogical interest forillustrating the dependence of the direction of the frictionforce on the initial translation/spin conditions.

As shown in Fig. 4, the data for x / xo versus xo for therebound of steel bearings on a ceramic floor can be dividedinto two linear regions, corresponding to collisions with andwithout sliding. Data points S 1 and S 2 in Fig. 4 correspondsto the collisions of steel bearings of diameter 1.28 and 2.50cm, respectively, launched on a hard wood surface after roll-ing along the horizontal track. The balls are launched withforward spin R oy / vox equal to the ratio R / R e ,

22 where Reis the effective radius of the sphere, defined as the distancefrom its center of mass to the line joining the points of con-tact with the track. It can easily be demonstrated that Re( R 2a2 /4)1/2, where a is the width of the track.22 The

data in the sliding region fits Eq. 25 if we let e0.78 and

0.07 in both cases. The data in the nonsliding region also

fit the predicted results if we set e0.78 in Eq. 27. Data S r corresponds to a steel bearing launched without initial spinwith the help of a spring-operated shooter, whereas data S s inFig. 4 corresponds to a steel bearing launched after beingstruck by an auxiliary cue ball. Again, the data are consistentwith theory using the above values of e and . For data

points S s , with e0.78 in Eq. 27, the data for the nonslid-

ing region lead to R oy / vox0.21, which is a reasonable

value.23 Experimental data S o corresponds to the rebound of a steel bearing launched with pure rolling motion along atube, that is, R oy / vox1. The data fit the expected curve

for zero friction forces with e0.78. The inset in Fig. 4shows the theoretical dependence for the correspondingcases. The ‘‘jump’’ that separates the linear regions occursbecause the static and kinetic friction coefficients differ fromeach other. When the calculations are performed with equal

values of the two friction coefficients, these discontinuitiesdisappear, but discontinuities in the slope remain at points of transition from collisions involving sliding at the instant inwhich the contact between the sphere and the surface ceasesto those not involving sliding at this instant.

The data for golf balls also agree with the -e modelunder our experimental conditions. From our data, no satis-factory estimates of the static coefficient of friction wereobtained because no clear discontinuity was obtained be-tween the lines fitting experimental data to the sliding andthe nonsliding regions. However, the data limited it to morethan 20% of the kinetic coefficient in all cases.

The linearity in the x / xo versus x o plots in Fig. 4 as well

as the tan versus tan plots in Fig. 3 suggests that the

assumption that both friction and restitution coefficients areconstant is reasonably valid under our experimental condi-tions. Also in agreement with theory, the ordinate of the x / x oversus xo graphs in the -e region is equal to 2 e regardless

of the initial spin conditions and the slope is (1

e)g / uo2 for vox R oy0 or (1e)g / u o

2 for vox R oy0. The consistency of the values of e and calcu-lated for different steel balls under different initial spin con-ditions support the idea that these coefficients can be treatedas constants.

C. The superball regime

The post-rebound spin is the quantity most sensitive tochanges in the impact regime. The values of R oy / u o , cal-

culated from x , x o , x*, xo* see Eq. 30 for a steel bearingbouncing without initial spin on a wood surface, are shownin Fig. 5. At low impact angles these values are consistentwith theory for a nonsliding rebound line a, R oy / u o5/7), whereas for relatively large impact angles, the -e

model Eq. 11 applies with e0.52 and 0.10 line c.

A discontinuity appears in the R oy / u o values near

18°. This discontinuity can be associated with the static

coefficient of friction o0.150.02 see curve d.

However, for the superball, the ratio R oy / uo circles in

Fig. 5 remains consistently equal to the value ( R oy / u o

1.07) from the e -e model Eq. 21 with e x

0.50 as canbe seen in Fig. 5 line b. This situation corresponds to gripbehavior. The transition from this grip regime to an ordinarysliding regime occurs at 76°, consistent with the values

e0.96, 0.95, depicted in curve e.The rebound of superballs is characterized by its large

elasticity restitution coefficient close to one and by the ap-pearance of large friction forces resulting in large post-impact spins. Prior data indicate that the response of super-balls cannot be satisfactorily described within the -e

model.9 To test the application of the e -e model, we compareits predictions with published data on post-rebound linear

Fig. 4. Comparison of experimental x / x o ratios for steel balls bounced on afloor with predicted values using e0.78 and 0.07 in Eqs. 25 and 27.

a experimental data S 1 : diameter d 1.28 cm, initially in pure rolling

motion along the horizontal track, R oy / vox1.60; b experimental data

S 2 : d 2.50 cm, initially in pure rolling motion, R oy / vox1.09; c ex-

perimental data S r : d 2.50 cm, R oy / vox0; d experimental data S s :

d 2.50 cm, R oy / vox0.21; e experimental data S o : steel bearing

(d 2.50 cm) launched with pure rolling motion with R oy / vox1.00. In-

set: expected plots for o .



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and angular velocities. The data of Ref. 9 corresponds toexperiments in which a golf ball and a superball bouncewithout initial spin on a hard surface. In contrast with ourexperiments, the horizontal velocity of the ball before thebounce was not constant and, hence, the ratio R y / vo was

used rather than R y / u o .

In Fig. 6 the data taken from Ref. 9 for R y / vo for a golf

ball filled circles and a superball circles are comparedwith the theoretical expectations from the nonsliding regimea, the e-e model with e x0.50 line b, and the idealresponse described by Garvin line c.6 The predictions forthe Amontons–Coulomb regime from the -e model Eq.

11 are represented as dotted lines using the and e values

reported in Ref. 9. For the superball e0.96, 1.0 line

f , and for the golf ball e0.90, 0.18 line d. Line

e corresponds to the values e0.90 and 0.22, a valuethat is representative of the static coefficient of friction for

the rebounds of the golf ball. The line g marked by arrowsin Fig. 6 corresponds to the variation of R oy / uo with theincident angle of the golf ball, assuming that a transitionfrom the -e sliding regime to the nonsliding regime occurs.As can be seen in Fig. 6, excellent agreement exists betweentheory and experimental data using the previously mentionedvalues of e, , and o .

The experimental data for the rebound of a superball on ahard surface reported in Ref. 9 are consistent with predic-tions of the e-e model Eq. 21 by using e x0.50, clearlyindicating that this model must be used for rebounds withlarge friction. The transition from this regime to a -e slid-ing regime can occur only at low values of and cannot beclearly seen in the data in Fig. 6. Similar considerations canbe obtained from the experimental values of ( R yv x)/ voxe x in Ref. 9.

All these results can be understood by assuming that thetransition from the ordinary sliding regime to a nonslidingregime occurs when the friction force reaches a value givenby the coefficient of static friction. In ordinary systems, o and the loss of sliding motion leads to a nonslidingregime. The behavior of superballs can be interpreted byassuming that o . In this case, grip-slip behavior is ob-served. The transition from this grip-slip regime to the ordi-nary sliding regime occurs when the friction force equals acritical value that depends on e x . In these circumstances, thenonsliding regime is absent. These considerations are consis-

tent with the value of the coefficients of static and dynamicfriction estimated here and those reported by Cross8,9 for therebound of superballs with hard surfaces. The static coeffi-cient in Ref. 8 was 0.520.04, while the dynamic coefficientestimated in Ref. 9 was about unity for comparable surfaces.Under our experimental conditions, the dynamic coefficientof friction for the superball on different surfaces was esti-mated to be between 0.85–0.90, also larger than the reportedstatic values.8

D. Some implications for sports

From our results, some brief observations concerning the

mechanics of ball sports can be made. Horizontal launchexperiments can be used to determine the mechanical char-acteristics of a tennis court, thus obtaining the coefficients of restitution and friction and eventually the conditions for slip-ping, no slipping, and grip-slip regimes of rebound.

For tennis it is useful to obtain large rebound angles: therebound angle depends on the materials for example, clayand grass and the conditions of the court, given by the val-ues of e and , but also on the angular velocity imparted to

the ball the lift. Figure 7 shows R oy / vox versus for e

0.80, close to the values reported by Cross9 for the reboundof tennis balls on hard surfaces. The external sliding regions

Fig. 5. Variation of R y / uo for balls bouncing on a wood surface. The balls

were projected horizontally without an initial spin at a constant horizontal

velocity of 1.000.05 m/s. Points: steel ball, diameter 2.50 cm; circles:

superball. a theoretical plots for a nonsliding regime, b e -e grip regime

with e x0.50; c -e sliding regime with e0.52, 0.10; d e

0.52, 0.15; and e e0.96, 0.95.

Fig. 6. Plots of R oy / vo versus the incident angle ( 90° ). Compari-

son of the data for the rebound of a golf ball points and a superball

circles taken from Ref. 6, and theoretical predictions using the nonsliding

condition a, the e-e model Eq. 21 with e x0.50 b and the Garvin

model c. Dotted lines correspond to the -e sliding regime Eq. 11 with

e0.90, 0.18 d, and e0.90 and 0.22 e and e0.96, 1.0 f .

Line g corresponds to the variation of R oy / uo with the incident angle

of the golf ball, assuming that a transition from the -e sliding regime tothe nonsliding regime occurs.



8/9

for which a -e sliding regime must be attained are a e

0.80, 0.40, and b e0.80, 0.10, and are limitedby transitions to the corresponding nonsliding regime con-tinuous lines and to a grip-slip regime with e x0.50 dottedlines. Figure 7 exhibits symmetry with respect to R oy / vox1. Obviously, a backspin must be imparted to theball to achieve large rebound angles, a condition frequentlydesired by tennis players.

Similar considerations can be applied to golf. Initially it isdesirable to hit the ball to a large distance. In this case it isconvenient to impart a forward spin to the ball because thefriction force arising in the rebound increases the linear ve-locity of the center of mass of the ball after the rebound.However, for hits in which the ball must be located withinthe ‘‘green,’’ it is convenient to impart a backspin; in thiscase the velocity of the center of mass of the ball is de-creased; that is, the ball covers a small distance after therebound.

In basketball and handball, the players move while bounc-ing the ball. In these cases it is desirable that the velocity of the center of mass of the ball remains equal before and afterthe bounce, or, equivalently, . This condition can beobtained by imparting a backspin to the ball with the fingersas described in Ref. 4. The magnitude of the backspin de-pends on the restitution and friction coefficients. In thesecases, however, the impact angle is usually low, and we ex-

pect that the ordinary nonsliding regime applies, that is, therebounds must fall into the nonsliding region of the diagramin Fig. 7.

In tennis it often is desirable to obtain rebounds in whichthe ball is displaced horizontally. In these cases, a spin aboutthe x axis should be imparted to the ball. If the nonsliding

regime holds, we obtain from Eq. 23 with e x0 that the

deviation angle is only a function of R ox / vo , and isindependent of the incident angle. The same result is ob-tained for grip-slip behavior. In contrast, if the -e slidingregime applies, Eq. 12 predicts that for a given value of R ox / vo , the deviation angle decreases as the incident angle

increases. As a general rule, obtaining a large deviation anglerequires a low angle of impact. Similar considerations applyfor handball, a sport in which lateral deviation effects areimportant.

V. DISCUSSION

The bounce of a ball on a horizontal surface can be de-scribed using two limiting models: the -e model, in whichthe friction forces are described in terms of the static anddynamic coefficients of friction, and the e -e model, in which

the friction forces are expressed in terms of horizontal coef-ficients of restitution. When the rebound takes place withoutsliding at the point of contact, both models lead to identicalpredictions.

The classic horizontal launch experiment provides an em-pirical way of distinguishing between the sliding, nonsliding,and grip regimes. The data indicate that the rebound of ordi-nary balls can be described in terms of the transition from asliding regime, satisfactorily described by the -e model, toa nonsliding regime. This situation occurs in the usual casein which o . The rebound of superballs, however, cor-responds to a grip behavior that can be satisfactorily de-scribed by the e-e model. This difference appears to be as-

sociated with the condition o

. The data suggest thatthere is a transition from the grip behavior to the ordinarysliding one while the nonsliding regime is entirely absent.

The horizontal launch experiment can be used to deter-mine the characteristics of courts in several sports. Frompedagogical purposes, the horizontal launch experiment canbe used to illustrate the laws of impact taking into accountinelasticity and friction. Additionally, the experiment dis-cussed in Sec. III is illustrative of the design of suitableexperiments for comparing theoretical models.

Note, that the scope of the models we have discussed islimited by the assumption of the constancy of the coefficientsof restitution and friction. The fit between theory and experi-ment suggests that this assumption is reasonable for the con-

ditions of moderate velocity and small surface deformationimposed here, in agreement with published data.7–9,13,15,17

The methodology can be used to study the variation of ande with parameters such as the velocity and the diameter. Inspite of these limitations, the models discussed in this papercan be considered as plausible and complementary descrip-tions of rebounds in three dimensions.

aElectronic mail: [email protected]. Metha and J. M. Luck, ‘‘Novel temporal behavior of a nonlinear dy-

namic system: The completely inelastic bouncing ball,’’ Phys. Rev. Lett.65, 393–396 1990.

2A. Metha and G. C. Barker, ‘‘The dynamics of sand,’’ Rep. Prog. Phys. 57,

383–416 1994.3A. Lorenz, C. Tuozzolo, and M. Y. Louge, ‘‘Measurements of impact

properties of small, nearly spherical particles,’’ Exp. Mech. 37 , 292–2981997.

4P. J. Brancazio, ‘‘Physics of basketball,’’Am. J. Phys. 49, 356–365 1981.5H. Brody, ‘‘That’s how the ball bounces,’’ Phys. Teach. 22, 494–497

1984.6R. Garwin, ‘‘Kinematics of ultraelastic rough ball,’’ Am. J. Phys. 37,

88–92 1969.7R. Cross, ‘‘The bounce of a ball,’’ Am. J. Phys. 67 , 222–227 1999.8R. Cross, ‘‘Measurements of the horizontal coefficient of restitution for a

superball and a tennis ball,’’ Am. J. Phys. 70, 482–489 2002.9R. Cross, ‘‘Grip-slip behavior of a bouncing ball,’’ Am. J. Phys. 70, 1093–

1102 2002.10N. Maw, J. R. Barber, and J. N. Fawcett, ‘‘The role of elastic tangential

compliance in oblique impact,’’ J. Lubr. Technol. 103 , 74–80 1981.

Fig. 7. The different rebound regimes for R ox0. Continuous lines corre-

spond to the transition from the -e sliding regime to the nonsliding one for

a e0.80, 0.40, and b e0.80, 0.18. Dotted lines correspond to

the transition from the e-e grip regime with e x0.50, to the -e sliding

regime with the values of the coefficients of friction and restitution in a.



9/9

11E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and

Rigid Bodies Cambridge University Press, Cambridge, 1904, p. 232.12J. B. Keller, ‘‘Impact with friction,’’ J. Appl. Mech. 53 , 1–3 1986.13R. M. Brach, ‘‘Friction, restitution, and energy loss in planar collisions,’’ J.

Appl. Mech. 51 , 164–170 1984.14T. R. Kane and D. A. Levinson, ‘‘An explicit solution of the general

two-body collision problem,’’ Comput. Mech. 2, 75–87 1987.15G. Barnes, ‘‘Study of collisions. Part I: A survey of the periodical litera-

ture,’’ Am. J. Phys. 26, 5–8 1958; ibid. ‘‘Study of collisions. Part II:

Survey of the textbooks,’’ Am. J. Phys. 26 , 9–12 1958.16G. Amontons, ‘‘De la resistance causeé dans les machines,’’ Memoires de

l’Academie Royale A, 275–282 1699; briefly described in J. Krim, ‘‘Re-

source Letter: FMMLS-1: Friction at macroscopic and microscopic length

scales,’’ Am. J. Phys. 70 , 890–896 2002.17H. L. Armstrong, ‘‘How dry friction really behaves,’’ Am. J. Phys. 53,

910–911 1985.18A. Doménech and M. T. Doménech, ‘‘Analysis of two-disc collisions,’’

Eur. J. Phys. 14 , 177–183 1993.19A. Doménech and M. T. Doménech, ‘‘Oblique impact of rolling spheres: A

generalization of billiard-ball collisions,’’ Rev. Mex. Fis. 44, 611–618

1998.20Strictly speaking, friction associated with the rotation of the sphere around

a vertical axis, resulting in a decrease of z , appears. This effect has been

neglected here Ref. 10.21R. A. Bachman, ‘‘Sphere rolling down a grooved track,’’ Am. J. Phys. 53,

765–767 1985.22For the case of pure rolling motion along a horizontal track, the term

R oy / u o equals the ratio of the effective radius of gyration and the radius

of the ball. See, for instance, D. E. Shaw and F. Wunderlich, ‘‘Study of the

slipping of a rolling sphere,’’ Am. J. Phys. 52, 997–1000 1984; R. L.

Chaplin and M. G. Miller, ‘‘Coefficient of friction for a sphere,’’ Am. J.

Phys. 52 , 1108–1111 1984.23For the case of the head-on impact of spheres along a horizontal track,

R oy / u o must vary from (5/2) S , where S is the coefficient of fric-

tion between the spheres, to R / Re , when the ball reaches the pure rolling

motion along the track. See A. Doménech and E. Casasús, ‘‘Frontal impact

of rolling spheres,’’ Phys. Educ. 26 , 186–189 1991.

Steam Turbine. Models of reciprocating steam engines from the first half of the 20th century are fairly common, but this turbine-type engine in the

collection at Hobart and William Smith Colleges is unique. A 400 W electric heater acting on the water in the boiler produces the jet of steam from the nozzle

at the right. The turbine and its blades are covered with wire mesh to protect inquiring fingers. A pulley, hidden behind the turbine disk, is used to take off the rotary motion. The apparatus is listed at $8.50 in the 1940 Central Scientific Company catalogue, and uses the same boiler and safety-valve mechanism

as the accompanying reciprocating steam engine model. Photograph and notes by Thomas B. Greenslade, Jr., Kenyon College


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