aa r X i v : . [ phy s i c s . c l a ss - ph ] J u l Delayed Rebounds in the Two-Ball Bounce Problem
Sean P. Bartz ∗ Dept. of Chemistry and Physics, Indiana State UniversityTerre Haute, IN 47809
July 31, 2020
Abstract
In the classroom demonstration of the two-ball drop, some conditions lead to a“delayed rebound effect,” with the second bounce of the upper ball higher than thefirst. This paper uses two models to explore the causes of this phenomenon. The classicindependent contact model (ICM) is reviewed for the first bounce, and extended semi-analytically to the second bounce in the perfectly elastic case. A dynamical modelbased on a linear dashpot force is studied numerically. The delayed rebound effect isfound for a range of parameters, most commonly in cases where the first bounce islower than the ICM prediction.
In a classic classroom demonstration of linear momentum conservation, a tennis ball is heldabove a basketball, and the two are simultaneously dropped to the floor. Surprisingly, thetennis ball rebounds much higher than the drop height. Textbook explanations suggest thatwhen the upper ball is much less massive than the lower ball, it rebounds at three times theimpact speed, bouncing to nine times the initial drop height [1, 2, 3].Taking this theoretical prediction to heart, I brought a ping-pong ball for an in-classdemonstration, alerting students to expect a bounce more impressive than the tennis ball’sdue to the larger mass disparity. The demo was a dud, with the ping-pong ball staying closeto the basketball on the first bounce. However, with the balls carefully aligned, the smallfirst bounce was followed by a noticeably higher second bounce. This paper will show thatthis “delayed rebound effect” is robust, and can be explained with a simple force model.The classic justification for the high bounce of the tennis ball assumes that the basketball-floor collision is independent of the basketball-tennis ball collision. Closer inspection indi-cates that the simplifying assumption of this independent contact model (ICM) is invalid –the lower ball often remains in contact with the floor when the two balls first make contact. ∗ [email protected] The classic textbook solution to the two-ball drop problem assumes independent, instanta-neous collisions between the balls and the floor. We review the ICM here to form a basis ofcomparison for the more-realistic dynamic model, and to introduce notation. We also extendthe model to a second bounce for the elastic case.Inelastic collisions are characterized by a coefficient of restitution, defined as the ratio ofthe relative velocity of the particles post-collision to their pre-collision relative velocity, ε = (cid:12)(cid:12)(cid:12)(cid:12) v ′ − v ′ v − v (cid:12)(cid:12)(cid:12)(cid:12) . (1)Here, primed velocities refer to the velocity just after the collision. Using this relation, alongwith conservation of momentum, the post-collision velocities are v ′ = v + µ ( v + ε ( v − v ))1 + µ (2) v ′ = µv + v + ε ( v − v )1 + µ , (3)where µ = m /m .For the stacked ball drop, we define z as the distance between the bottom of the lowerball and the floor and z as the height of the upper ball, as measured from the top of thelower ball when it rests on the floor (assuming the ball does not compress). See Figure 1 foran illustration. This coordinate definition, where z = 0 and z = 0 do not occur at the samephysical point, explicitly removes reference to the radii of the balls, allowing us to extractuniversal behavior. The balls are assumed to move in the vertical direction only. The resultsof the ICM from [9] can be applied to our coordinate definition by taking the limit that theball radii go to zero. 2 z ( t ) m m z ( t )Figure 1: Coordinate definitions: z is measured from the floor to the bottom of the lowerball. The coordinate for the upper ball, z , is measured from the top of the lower ball whenit rests on the floor without compression.We define the drop height h = z (0) and the gap ∆ h = z (0) − z (0). Using conservationof energy, we can extend the results of [9] to calculate the maximum height of the upper ballafter the first bounce h = h (( µ − ε (2 + ε )) + ∆ h (1 + µ )( µ − ǫ )(1 + µ ) . (4) Analytical description of subsequent bounces is possible, but presents practical challenges.The primary challenge is the lower ball bouncing on the floor one or several times beforecontacting the upper ball a second time. The trajectory of the lower ball must thereforebe described in a piecewise fashion, precluding a general algebraic solution for the secondcollision. Because a numerical solution is necessary in any case, we do not exhaustivelyexamine the second-bounce effect in the ICM. Rather, we examine perfectly elastic collisionswith no initial gap between the balls as an illustrative example.In this section, we take t = 0 to be the time of the first collision. The velocity of aperfectly elastic single bouncing ball is a sawtooth function, which can be written v ( t ) = 2 v ′ π arctan (cid:18) cot πtt b (cid:19) , (5)where where v ′ is the post-collision velocity (2) with ε = 1 and v = √ gh . The timescale t b = 2 v ′ /g is the time a ball will spend between bounces. This model is valid only for µ < /
3, as the post-collision velocity (2) is negative for greater values, making (5) invalid.Conservation of energy gives a direct calculation of the vertical position from the velocity z ( t ) = v ′ g − v ′ πg arctan (cid:18) cot πtt b (cid:19) . (6)3he second collision occurs when z ( t c ) = z ( t c ), with t c the time of the second collisionbetween the two balls. This reduces to the simple expression v ′ − v ( t c ) = v ′ − v ( t c ) . (7)Inserting (5) and using free-fall kinematics for the upper ball yields v ′ − ( v ′ − gt c ) = v ′ − v ′ π arctan (cid:18) cot πt c t b (cid:19) . (8)This transcendental equation is solved numerically for t c , from which the position and veloc-ities of the balls at their second collision are calculated. The velocities following the secondcollision are calculated via (2) and (3), and the height of the second bounce is calculatedusing free-fall kinematics.The heights for the first and second bounce for various values of µ are shown in Figure2, normalized by the initial drop height. In the limit µ →
0, the well known first-bouncevelocity ratio of 3 is recovered, which leads to the ball bouncing to 9 times its drop height.The second bounce approaches a limit of 25.While this simple case of the ICM shows a delayed rebound effect for a range of µ values,the quantitative predictions do not comport with informal observations. Namely, high secondbounces are generally observed in situations where the first bounce is not large compared tothe drop height. However, the ICM results show that the existence of the delayed reboundeffect does not depend on the details of the interaction between the balls, and suggest that itis more prominent at smaller µ values, matching observations and the model results reportedin the following sections. Following the work in [9] we use a linear dashpot force as a simplified approximation of theHertz contact force between viscoelastic spheres [5, 10]. The force is F ij = − min[0 , − kξ ij − γ j ˙ ξ ij ] , (9)where the mutual compression is defined by ξ ij = Θ( − z i + z j ) , (10)where z i are the vertical positions of the balls as defined in Figure 1 and Θ( x ) is the Heavisidestep function. The floor is denoted by index 0, while the lower and upper balls are labeledby index 1 and 2, respectively. The force definition (9) ensures that the force is alwaysrepulsive.To simplify the analysis, the restoration constant k is taken to be the same for bothballs, and the coefficient of restitution is set by varying the damping constants γ i . The ballsare considered to be in contact when the force between them is nonzero, rather than whenthe mutual compression is nonzero. In the case of γ = 0, the collision is perfectly elastic,and the ICM is reproduced when the initial gap ∆ h is large enough that the collisions areindependent. 4 irst BounceSecond Bounce0.00 0.05 0.10 0.15 0.20 0.25 0.300510152025 μ B oun ce h e i gh t D r oph e i gh t Figure 2: The bounce height of the upper ball, normalized by the initial drop height of theupper ball, shows the “delayed rebound effect” is present in the ICM for perfectly elasticcollisions. In the limit µ →
0, the well-known ratio of 9 is recovered for the first bounce, andthe second bounce approaches a limit of 25.The equations of motion are m ¨ z = − m g + F − F (11) m ¨ z = − m g + F (12)To isolate the parameters of physical importance, the equations of motion are written in adimensionless fashion. The equations are non-dimensionalized via the substitutions z i = x c X i and t = t c τ, where x c = m g/k and t c = p m /k . The equations become X ′′ = − f − f (13) X ′′ = − f /µ, (14)where ( ′ ) indicates a derivative with respect to τ . The dimensionless forces are f ij = − min (cid:2) , Ξ ij + 2 ζ j Ξ ′ ij (cid:3) (15)where ζ i = γ i / √ m k are the damping ratios and Ξ ij = Θ( − X i + X j ) is the dimensionsless5
15 316 317 318 319 320 321−2500250500 D i m e n s i o n l e ss p o s i t i o n X X
315 316 317 318 319 320 321τ0102030 D i m e n s i o n l e ss F o r c e f /10f Figure 3: The trajectories of the balls and the force curves during the first collision are shownfor a representative case where τ d /τ f = 0 . . In this case, µ = 0 . ε = ε = 0 . ε = exp " − ζ p ζ − ζ + p ζ − ζ − p ζ − ! (16) ε = exp − ζ q ζ − µ µ ln ζ + q ζ − µ µ ζ − q ζ − µ µ , (17)with the principal branch cut used for the natural logarithm when the argument becomescomplex. The expressions can be made manifestly real-valued in a piecewise fashion, asshown in Appendix A. The problem is now defined in terms of the three dimensionlessconstants µ, ε , ε , and two initial conditions X (0) , X (0). The duration of contact between the lower ball and the floor is calculated in the adiabaticapproximation wherein the force from the upper ball has negligible effect [11]. The resulting6xpression in dimensionless units is τ f = √ − ζ (cid:18) π − arctan ζ √ − ζ − ζ (cid:19) for ζ < √ − √ − ζ arctan ζ √ − ζ − ζ for √ < ζ < − √ ζ − artanh ζ √ ζ − − ζ for ζ > . (18)This expression is exact in the case of independent collisions, or in the limit µ →
0. In othersituations, the actual floor contact time will be longer. The value of τ f is used to determineif the collisions are independent. The time interval between the time X = 0 and X = 0 τ d = √ (cid:16)p X (0) − p X (0) (cid:17) . (19)This is interpreted as the time at which the balls would collide if they did not compress orbounce. If τ d > τ f , then the collisions are independent. Otherwise, the lower ball will stillbe in contact with the floor when the upper ball collides with it.Figure 3 shows the trajectories of the two balls and the force curves during the lowerball’s first collision with the floor. In this case, τ d /τ f = 0 . , but the first contact of theballs occurs later than 0.01 of the way between initial and final contact of the lower ballwith the floor, due to the compression of the lower ball. (See Appendix B for details of thetrajectories during this phase of motion.) The compression of the balls is illustrated by thefact that both X and X become negative in Figure 3.For given values of ε , ε , and µ , the bounce height ratios are found to be the same if τ d /τ f is kept fixed. For different drop heights, the timing of the collision and the overall scaleof the forces will change, but the overall form remains the same when τ d /τ f is held constant.Thus, the situation is described entirely by three parameters ( ε , ε , µ ) and a single initialcondition ( τ d /τ f ). Most analysis of the two-ball drop problem focuses on the first bounce [12] , taken here toencompass the time from which the balls are released until the upper ball reaches its nextlocal maximum in height. The collisions of interest for the first bounce include the firstcollision between the lower ball and the floor, and one or several collisions between the twoballs. We extend our analysis to include the second bounce of the upper ball. The lower ballwill contact the floor one or more times before the balls collide for the second time, and thesecond collision may or may not occur with the lower ball in contact with the floor.To simplify the analysis, we set ε = ε throughout the rest of this work. In general,these coefficients of restitution may differ, but the results are shown to be qualitativelysimilar when ζ = ζ instead [9]. The ball parameters ε, µ and the initial condition τ d /τ f were each divided into 50 increments, for 50 parameter combinations. The ranges weretaken as ε ∈ (0 . , µ ∈ (10 − ,
1) and τ d /τ f ∈ (10 − , ε = 0 . Analysis of the forces (15) from simulations with simultaneous contacts reveals multiplecontacts between the two balls under a variety of conditions, consistent with the analysis in[9]. Figure 3 shows a case with four contacts between the balls, occurring entirely while thelower ball is in contact with the floor. Figure 4 shows a case where a second contact occursjust after the lower ball has left the floor.The number of contacts between the two balls depends sensitively upon the ball param-eters and initial condition. The count varies between one and twelve for the representativedata shown in Figure 5, with ε = 0 . µ and τ d /τ f are small, so the plot axes are scaled logarithmically to show this detail, and the rangeof τ d /τ f is expanded to (10 − ,
315 316 317 318 319 320 3210500 D i m e n s i o n l e ss p o s i t i o n X X
315 316 317 318 319 320 321τ0102030 D i m e n s i o n l e ss F o r c e f /10f Figure 4: In this simulation, µ = 0 . , τ d /τ f = 0 . , and ε = ε = 0 .
9. One of the collisionsbetween the balls occurs just after the lower ball leaves the floor.Because contact between the two balls is defined by f = 0, instead of by the balls’positions, experimental setups that measure position only will not confirm or refute thedetails of the collisions presented here [16]. Piezoelectric sensors placed between the ballsand on the floor present a more promising experimental setup, but the situations studied in[17] do not show clear evidence for multiple contacts. However, the particular measurementsin that paper do not conflict with our simulations, as they do not match conditions predicted8o exhibit multiple collisions. Future experiments targeting the parameters that predictmultiple contacts would help validate the use of this model. −4 −3 −2 −1 μ10 −4 −3 −2 −1 τ d / τ f Number of contacts with ε = 0.816 C o n t a c t s Figure 5: The number of contacts between the two balls during the first bounce with atypical coefficient of restitution ε = 0 . While the details of the forces during the collision are interesting, the post-collision motionof the balls is more readily observable in an experimental setting. The analysis presentedhere focuses on the motion of the upper ball in part because the lower ball’s motion is largelyunaffected by the collision for small mass ratios. We focus on the maximum height of thisball after each bounce rather than the relative velocity of the balls because the height ofthe second bounce is influenced by the post-collision velocity and the height of the secondcollision.When the initial gap between the balls is small ( τ d /τ f ≪ τ d /τ f >
1, asexpected when collisions are independent. (Note that τ d /τ f ≫ −4 −3 −2 −1 μ10 −4 −3 −2 −1 τ d / τ f First bounce height with ε = 0.816
Figure 6: Comparison to Figure 5 shows no clear relationship between number of contactsand the height the upper ball reaches on its first bounce.height is greater than that of the ICM. The ICM is overperformed when ε is small and µ ison the large end of the range studied, as shown in Figure 7. These conditions lead to smallbounce heights, below the initial drop height, in any case.The ICM closely approximates post-collision behavior for a variety of cases where thecollisions are not truly independent, as τ d /τ f <
1. For τ d /τ f ≈ .
7, there is deviation fromthe ICM for some regions of the parameter space shown in Figure 8, but most situationsare well-approximated by the simple model. Plots of larger values of τ d /τ f are not shownbecause they do not exhibit noticeable contrast, as all results are quite close to the ICMvalue. For example, the bounce heights are all within 5% of the ICM limit for τ d /τ f > . In this section, we analyze the height of the upper ball on the second bounce, specificallystudying the delayed rebound effect with the second bounce higher than the first. Figure9 shows a comparison of the second bounce height to the height of the first bounce using ε = 0 .
816 as an illustrative example. This plot shows that most cases of the delayed reboundeffect occur for small initial drop gaps and small mass ratios, Qualitatively resembling theICM result in Figure 2. The region expands as ε increases. As expected, the small mass-ratio limits are approximately achieved in the elastic case ( ε = 1), when the initial collisions10 .00 0.05 0.10 0.15 0.20 0.25 0.30μ0.50.60.70.80.91.0 ε Bounce height ratio with τ d /τ f = 0.4141 s t b o un c e / I C M p r e d i c t i o n Figure 7: For moderate initial gaps, as shown here, there are some combinations of ε, µ thatresult in bounces lower than the ICM prediction, while others result in bounces higher thanthe ICM. The ratios plotted range from 0.49 to 1.48.between the floor and the two balls are independent ( τ d /τ f = 1).In Figure 10, the first and second bounce heights are compared. It is evident that thedelayed rebound effect is most prominent in cases where the first bounce is low, often lowerthan the initial drop height. The highest second bounces on an absolute scale occur whenthe first bounce is also high. Some of these do slightly exceed the first bounce, but thesecases are relatively few.Visual inspection gives a general sense of the causes of the delayed rebound effect thatis confirmed by systematic comparison. For the parameter ranges studied, 12.3% of combi-nations resulted in the delayed rebound effect. Of these , 93% occurred in cases where thefirst bounce was lower than the ICM prediction. However, having a first bounce lower thanthe ICM prediction was not highly predictive; only 22% of these cases have a higher secondbounce.Typically, the lower ball is not in contact with the floor when the balls collide for thesecond bounce, and the balls make a single contact with each other. When the lower ball isin contact with the floor, multiple contacts between the balls are possible, in a manner thatis qualitatively similar to the first bounce. These cases lead to a noticeably lower secondbounce than others in nearby parameter space.11 .00 0.05 0.10 0.15 0.20 0.25 0.30μ0.50.60.70.80.91.0 ε Bounce height ratio with τ d /τ f = 0.6969 s t b o un c e / I C M p r e d i c t i o n Figure 8: The results of the simulation begin to to converge to the ICM result for a widerange of ε, µ , despite the collisions not being independent, with τ d /τ f <
1. The ratios plottedrange from 0.87 to 1.27.
We show that the “delayed rebound effect,” where the second bounce of two aligned balls ishigher than the first, is present in both a semi-analytic independent contact model (ICM)and the numerical solutions to a linear dashpot force between the balls. The effect is ismost prominent when the upper ball has a much smaller mass than the lower ball, andthe distance gap between the balls is small when they are released. Typically, the finiteduration of the collisions leads the first bounce to be smaller than predicted by the ICM, sothe expected high bounce does not come until the second bounce. This is consistent withthe informal observation that inspired this study: namely, that the delayed rebound effectis more commonly seen in ping-pong ball–basketball collisions than in tennis ball–basketballcollisions.Rather than focus on the parameters of a few particular sports balls, we look for universalbehavior. The relevant ball parameters are reduced to the mass ratio µ = m /m and thecoefficient of restitution ε , assumed to be the same for both balls. The initial conditionsof drop height and gap between the balls is reduced to a single parameter τ d /τ f , whichcharacterizes the time between the lower ball reaching the floor and the two balls makingfirst contact. This approach represents a simplification over previous studies, and can begeneralized to more than two balls.We examined the details of the first bounce collisions, finding multiple contacts be-12 .00 0.05 0.10 0.15 0.20 0.25 0.30μ0.00.20.40.60.81.0 τ d / τ f Bounce height ratio with ε = 0.816 n d b o un c e / s t b o un c e Figure 9: Comparison of the second bounce height to the first bounce height. Red pointsindicate situations in which the second bounce is higher than the first.tween the balls in some cases. However, the multiple impacts did not correlate with thepost-collision dynamics of the balls. For the first bounce, we found that small initial gapsresulted in bounces that were smaller than the ICM prediction, while the ICM is approx-imately correct for larger initial gaps, despite the overlapping collisions. This correspondsto experimental and theoretical results, albeit with a greatly simplified description of thecontact force than in [16]. Direct comparison to realistic ball parameters for both first andsecond bounces is reserved for future work.While [9] suggests that the linear dashpot model gives qualitatively similar results tosimulations using the Hertz force, [18] finds qualitatively different results between theseforces when studying chain collisions. Thus, it may be worthwhile to repeat this study witha Hertzian contact force, more appropriate for viscoelastic spheres.A thorough experimental study of the delayed rebound effect requires a mechanism toconstrain the interacting particles to a single dimension. Precisely aligned spheres, as usedin experimental studies of the first bounce, are unlikely to remain aligned for a secondbounce. Low friction carts on an inclined track, with springs for repulsion, may be a usefulmodel, although it might be difficult to achieve the range of mass ratios seen in ball dropexperiments. 13 .00 0.05 0.10 0.15 0.20 0.25 0.30μ0.00.20.40.60.81.0 τ d / τ f First bounce height with ε = 0.816 s t b o un c e / ( d r o p h e i g h t + g a p ) (a) τ d / τ f Second bounce height with ε = 0.816 n d b o un c e / ( d r o p h e i g h t + g a p ) (b) Figure 10: Comparison of bounce heights to the initial height for a representative value ofthe coefficient of restitution. The plot in (a) shows that the first bounce exceeds the dropheight in most regions of parameter space. In (b), we see more detailed structure for thesecond bounce.
Acknowledgments
Thanks to Joe West for suggesting that my observation of my apparently failed classroomdemo could be turned into a paper. Numerical simulations were performed using NumPyand SciPy. The second bounce of the ICM was analyzed using Mathematica.The two-ball drop problem was the first physics problem that ever fascinated me, begin-ning with a demo at an IUPUI Saturday science program for high school students. Thanks tothe Macalester students who didn’t hold it against me when my overly-enthusiastic renditionof this demo resulted in a broken light bulb falling precariously close to them. Thank youto the Indiana State students for their patience as I repeated the seemingly unremarkableping-pong version of the demo. Maybe the reason for my fascination is now evident.I want to show my appreciation to my dog, Luna, who faithfully sat at my feet duringthe researching and writing of this paper during the COVID-19 quarantine. Good girl.
A Explicitly real coefficients of restitution
The expressions for the coefficients of restitution in (16,17) are both real-valued, but re-quire the handling of the natural logarithm of complex numbers. The following piecewise14xpressions are explicitly real-valued, and match the expressions in the main text. ε = exp (cid:20) − ζ √ − ζ (cid:18) π − arctan ζ √ − ζ − ζ (cid:19)(cid:21) for ζ < √ exp (cid:20) ζ √ − ζ arctan ζ √ − ζ − ζ (cid:21) for √ < ζ < (cid:20) − ζ √ ζ − ln (cid:18) ζ + √ ζ − ζ − √ ζ − (cid:19)(cid:21) for ζ > . (20) ε = exp (cid:20) − ζ √ µ µ − ζ (cid:18) π − arctan ζ √ (1+ µ )( µ − ζ (1+ µ )) µ − ζ (1+ µ ) (cid:19)(cid:21) for ζ < q µ µ ) exp (cid:20) ζ √ µ µ − ζ arctan ζ √ (1+ µ )( µ − ζ (1+ µ )) µ − ζ (1+ µ ) (cid:21) for q µ µ ) < ζ < q µ µ exp (cid:20) − ζ √ ζ − µ µ ln (cid:18) ζ + √ ζ − µ µ ζ − √ ζ − µ µ (cid:19)(cid:21) for ζ > q µ µ . (21) B Tra jectory of lower ball in contact with the floor
While the lower ball is in contact with the floor, its position is the trajectory of a a dampedharmonic oscillator X ( τ ) = 1cos p cos (cid:18) τ q − ζ + p (cid:19) e − ζ τ − , (22)where τ = 0 is redefined as the time at which the lower ball first contacts the floor, and p = arctan ζ + √ H p − ζ . (23)Here, H = h/x c is the dimensionless drop height.During this motion, the upper ball is in free fall. When the lower ball hits the ground,the position of the lower ball is X (0) = ∆ H , the initial gap in dimensionless coordinates.The time-dependent position is X ( τ ) = ∆ H − √ Hτ − τ . (24)Solving for the time and position of collision requires solving a transcendental equation,assuming τ d < τ f , as discussed in Section 3.1. Because of this, there is little practical reasonto prefer this method over the computational simulations performed. In either case, it isshown that the actual time time to collision is greater than τ d . Equation (22) assumes anadiabatic approximation, where the path of the lower ball is unaffected by collision(s) withthe upper ball. The expression becomes exact in the limit µ → τ d > τ f .15 eferences [1] W. R. Mellen, “Superball Rebound Projectiles,” American Journal of Physics no. 9, (Sept., 1968) 845–845.[2] W. G. Harter, “Velocity Amplification in Collision Experiments Involving Superballs,” American Journal of Physics no. 6, (June, 1971) 656–663.[3] F. Herrmann and P. Schmlzle, “Simple explanation of a well-known collisionexperiment,” American Journal of Physics no. 8, (Aug., 1981) 761–764.[4] J. D. Kerwin, “Velocity, Momentum, and Energy Transmissions in Chain Collisions,” American Journal of Physics no. 8, (Aug., 1972) 1152–1158.[5] P. Patr´ıcio, “The Hertz contact in chain elastic collisions,” American Journal of Physics no. 12, (Nov., 2004) 1488–1491, arXiv:0402036 [physics.ed-ph] .[6] M. Kire, “Astroblaster–a fascinating game of multi-ball collisions,” Physics Education no. 2, (Feb., 2009) 159–164.[7] M. Gharib, A. Celik, and Y. Hurmuzlu, “Shock Absorption Using Linear ParticleChains With Multiple Impacts,” Journal of Applied Mechanics no. 3, (May, 2011) 031005.[8] B. Ricardo and P. Lee, “Maximizing kinetic energy transfer in one-dimensionalmany-body collisions,” European Journal of Physics no. 2, (Feb., 2015) 025013.[9] Mller, Patric and T. Pschel, “Two-ball problem revisited: Limitations of event-drivenmodeling,” Physical Review E no. 4, (Apr., 2011) 041304, arXiv:1009.6153 [physics.class-ph] .[10] D. Gugan, “Inelastic collision and the Hertz theory of impact,” American Journal of Physics no. 10, (Sept., 2000) 920–924.[11] T. Schwager and T. Pschel, “Coefficient of restitution and lineardashpot modelrevisited,” Granular Matter no. 6, (Nov., 2007) 465–469, arXiv:0701278 [cond-mat.soft] .[12] J.-H. Ee and J. Lee, “Magic mass ratios of complete energy-momentum transfer inone-dimensional elastic three-body collisions,” American Journal of Physics no. 2, (Jan., 2015) 110–120, arXiv:1310.5200 [physics.class-ph] .[13] S. Bartz, “Data and visualizations.” https://seanbartz.com/research/data-and-visualizations/ .[14] A. C. Hindmarsh, “ODEPACK, A Systematized Collection of ODE Solvers,” IMACSTransactions on Scientific Computation (1983) 55–64.1615] L. R. Petzold, “Automatic selection of methods for solving stiff and nonstiff systems ofordinary differential equations,” SIAM Journal on Scientific and Statistical Computing no. 1, (1983) 136–148.[16] Y. Berdeni, A. Champneys, and R. Szalai, “The two-ball bounce problem,” Proc. R. Soc. Lond. A no. 2179, (July, 2015) 20150286.[17] R. Cross, “Vertical bounce of two vertically aligned balls,”
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