In a classic classroom demonstration of linear momentum conservation, a tennis ball is held above a basketball, and the two are simultaneously dropped to the floor. Surprisingly, the tennis ball rebounds much higher than the drop height. Textbook explanations suggest that when the upper ball is much less massive than the lower ball, it rebounds at three times the impact 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-class demonstration, alerting students to expect a bounce more impressive than the tennis ball's due to the larger mass disparity. The demo was a dud, with the ping-pong ball staying close to the basketball on the first bounce. However, with the balls carefully aligned, the small first bounce was followed by a noticeably higher second bounce. This paper will show that this "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 basketballfloor collision is independent of the basketball-tennis ball collision. Closer inspection indicates that the simplifying assumption of this independent contact model (ICM) is invalidthe lower ball often remains in contact with the floor when the two balls first make contact.
Interestingly, ICM predictions are fairly accurate in many cases when the collisions are not truly independent. However, when the initial separation between the two balls is small, the final velocities can differ greatly from the ICM prediction, particularly when the balls are allowed to bounce more than once.
Impact mechanics is relevant in a variety of fields including engineering, granular materials, and molecular dynamics. Study of this simple toy model may inform simulations in these practical applications by demarcating the conditions under which the ICM model is useful, and where more detailed simulations are required to accurately reflect macroscopic effects. Detailed study of two-particle interactions can also be extended to chain collisions of multiple particles in a line [4, 5, 6, 7, 8] .
The organization of this paper is as follows: the independent contact model is described in section 2. Section 3 develops a dynamic description of the system using a linear dashpot model. Section 4 uses the linear dashpot force to examine the details of the collisions in the first bounce, and examines the rebound height of the upper ball for both the first and second bounce.
The classic textbook solution to the two-ball drop problem assumes independent, instantaneous collisions between the balls and the floor. We review the ICM here to form a basis of comparison for the more-realistic dynamic model, and to introduce notation. We also extend the model to a second bounce for the elastic case.
Inelastic collisions are characterized by a coefficient of restitution, defined as the ratio of the relative velocity of the particles post-collision to their pre-collision relative velocity,
Here, primed velocities refer to the velocity just after the collision. Using this relation, along with conservation of momentum, the post-collision velocities are
where µ = m 2 /m 1 . For the stacked ball drop, we define z 1 as the distance between the bottom of the lower ball and the floor and z 2 as the height of the upper ball, as measured from the top of the lower ball when it rests on the floor (assuming the ball does not compress). See Figure 1 for an illustration. This coordinate definition, where z 1 = 0 and z 2 = 0 do not occur at the same physical point, explicitly removes reference to the radii of the balls, allowing us to extract universal behavior. The balls are assumed to move in the vertical direction only. The results of the ICM from [9] can be applied to our coordinate definition by taking the limit that the ball radii go to zero. We define the drop height h = z 1 (0) and the gap ∆h = z 2 (0) − z 1 (0). Using conservation of energy, we can extend the results of [9] to calculate the maximum height of the upper ball after the first bounce
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 before contacting the upper ball a second time. The trajectory of the lower ball must therefore be described in a piecewise fashion, precluding a general algebraic solution for the second collision. Because a numerical solution is necessary in any case, we do not exhaustively examine the second-bounce effect in the ICM. Rather, we examine perfectly elastic collisions with 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 a perfectly elastic single bouncing ball is a sawtooth function, which can be written
where where v ′ 1 is the post-collision velocity (2) with ε = 1 and v 1 = √ 2gh. The time scale t b = 2v ′ 1 /g is the time a ball will spend between bounces. This model is valid only for µ < 1/3, 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
The second collision occurs when z 2 (t c ) = z 1 (t c ), with t c the time of the second collision between the two balls. This reduces to the simple expression
Inserting (5) and using free-fall kinematics for the upper ball yields
This transcendental equation is solved numerically for t c , from which the position and velocities of the balls at their second collision are calculated. The velocities following the second collision are calculated via (2) and (3), and the height of the second bounce is calculated using free-fall kinematics.
The heights for the first and second bounce for various values of µ are shown in Figure  2 , normalized by the initial drop height. In the limit µ → 0, the well known first-bounce velocity 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 second bounces are generally observed in situations where the first bounce is not large compared to the drop height. However, the ICM results show that the existence of the delayed rebound effect does not depend on the details of the interaction between the balls, and suggest that it is more prominent at smaller µ values, matching observations and the model results reported in the following sections.
Following the work in [9] we use a linear dashpot force as a simplified approximation of the Hertz contact force between viscoelastic spheres [5, 10] . The force is
where the mutual compression is defined by
where z i are the vertical positions of the balls as defined in Figure 1 and Θ(x) is the Heaviside step function. The floor is denoted by index 0, while the lower and upper balls are labeled by index 1 and 2, respectively. The force definition (9) ensures that the force is always repulsive.
To simplify the analysis, the restoration constant k is taken to be the same for both balls, and the coefficient of restitution is set by varying the damping constants γ i . The balls are considered to be in contact when the force between them is nonzero, rather than when the 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 are independent. Figure 2 : The bounce height of the upper ball, normalized by the initial drop height of the upper ball, shows the "delayed rebound effect" is present in the ICM for perfectly elastic collisions. In the limit µ → 0, the well-known ratio of 9 is recovered for the first bounce, and the second bounce approaches a limit of 25.
The equations of motion are
To isolate the parameters of physical importance, the equations of motion are written in a dimensionless fashion. The equations are non-dimensionalized via the substitutions z i = x c X i and t = t c τ, where x c = m 1 g/k and t c = m 1 /k. The equations become
where ( ′ ) indicates a derivative with respect to τ . The dimensionless forces are
where mutual compression. The damping ratios are related to the coefficients of restitution [11] via
with the principal branch cut used for the natural logarithm when the argument becomes complex. The expressions can be made manifestly real-valued in a piecewise fashion, as shown in Appendix A. The problem is now defined in terms of the three dimensionless constants µ, ε 1 , ε 2 , and two initial conditions X 1 (0), X 2 (0).
The duration of contact between the lower ball and the floor is calculated in the adiabatic approximation wherein the force from the upper ball has negligible effect [11] . The resulting expression in dimensionless units is
This expression is exact in the case of independent collisions, or in the limit µ → 0. In other situations, the actual floor contact time will be longer. The value of τ f is used to determine if the collisions are independent. The time interval between the time X 1 = 0 and X 2 = 0
This is interpreted as the time at which the balls would collide if they did not compress or bounce. If τ d > τ f , then the collisions are independent. Otherwise, the lower ball will still be 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 lower ball's first collision with the floor. In this case, τ d /τ f = 0.01, but the first contact of the balls occurs later than 0.01 of the way between initial and final contact of the lower ball with the floor, due to the compression of the lower ball. (See Appendix B for details of the trajectories during this phase of motion.) The compression of the balls is illustrated by the fact that both X 1 and X 2 become negative in Figure 3 .
For given values of ε 1 , ε 2 , 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 scale of 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 1 , ε 2 , µ ) and a single initial condition (τ d /τ f ).
Most analysis of the two-ball drop problem focuses on the first bounce [12] , taken here to encompass the time from which the balls are released until the upper ball reaches its next local maximum in height. The collisions of interest for the first bounce include the first collision between the lower ball and the floor, and one or several collisions between the two balls. We extend our analysis to include the second bounce of the upper ball. The lower ball will contact the floor one or more times before the balls collide for the second time, and the second collision may or may not occur with the lower ball in contact with the floor.
To simplify the analysis, we set ε 1 = ε 2 throughout the rest of this work. In general, these coefficients of restitution may differ, but the results are shown to be qualitatively similar when ζ 1 = ζ 2 instead [9] . The ball parameters ε, µ and the initial condition τ d /τ f were each divided into 50 increments, for 50 3 parameter combinations. The ranges were taken as ε ∈ (0.5, 1), µ ∈ (10 −2 , 1) and τ d /τ f ∈ (10 −4 , 1). For the sake of comparison, the plots shown here focus primarily on the representative value of ε = 0.816, but animated visualizations of all data are available [13] . We use a standard numerical ordinary differential equation solver [14, 15] to integrate the initial value problem (13, 14) with the balls released from rest.
Analysis of the forces (15) from simulations with simultaneous contacts reveals multiple contacts 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 the lower ball is in contact with the floor. Figure 4 shows a case where a second contact occurs just after the lower ball has left the floor.
The number of contacts between the two balls depends sensitively upon the ball parameters and initial condition. The count varies between one and twelve for the representative data shown in Figure 5 , with ε = 0.816. Multiple contacts are mostly found when µ and τ d /τ f are small, so the plot axes are scaled logarithmically to show this detail, and the range of τ d /τ f is expanded to (10 −4 , 1). Because contact between the two balls is defined by f 2 = 0, instead of by the balls' positions, experimental setups that measure position only will not confirm or refute the details of the collisions presented here [16] . Piezoelectric sensors placed between the balls and 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 measurements in that paper do not conflict with our simulations, as they do not match conditions predicted to exhibit multiple collisions. Future experiments targeting the parameters that predict multiple contacts would help validate the use of this model. 
While the details of the forces during the collision are interesting, the post-collision motion of the balls is more readily observable in an experimental setting. The analysis presented here focuses on the motion of the upper ball in part because the lower ball's motion is largely unaffected by the collision for small mass ratios. We focus on the maximum height of this ball after each bounce rather than the relative velocity of the balls because the height of the second bounce is influenced by the post-collision velocity and the height of the second collision.
When the initial gap between the balls is small (τ d /τ f ≪ 1), the height of the first bounce is less than the ICM prediction (4). The ICM limit is recovered in the case τ d /τ f > 1, as expected when collisions are independent. (Note that τ d /τ f ≫ 1 results in collisions high above the floor. In this case the post-collision velocities will match the ICM, but the heights will not due to the height of the collision. These situations are not explored here.)
Intermediate gaps show more detailed structure. For some combinations of parameters, the bounce height remains below the ICM prediction, while for others the simulated bounce height is greater than that of the ICM. The ICM is overperformed when ε is small and µ is on the large end of the range studied, as shown in Figure 7 . These conditions lead to small bounce heights, below the initial drop height, in any case. The ICM closely approximates post-collision behavior for a variety of cases where the collisions are not truly independent, as τ d /τ f < 1. For τ d /τ f ≈ 0.7, there is deviation from the ICM for some regions of the parameter space shown in Figure 8 , but most situations are well-approximated by the simple model. Plots of larger values of τ d /τ f are not shown because they do not exhibit noticeable contrast, as all results are quite close to the ICM value. For example, the bounce heights are all within 5% of the ICM limit for τ d /τ f > 0.85 for the range of ball parameters studied. This unexpected success of the ICM is confirmed by experimental measurements [16, 17] .
In this section, we analyze the height of the upper ball on the second bounce, specifically studying the delayed rebound effect with the second bounce higher than the first. Figure  9 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 rebound effect occur for small initial drop gaps and small mass ratios, Qualitatively resembling the ICM result in Figure 2 . The region expands as ε increases. As expected, the small massratio limits are approximately achieved in the elastic case (ε = 1), when the initial collisions 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 the delayed rebound effect is most prominent in cases where the first bounce is low, often lower than the initial drop height. The highest second bounces on an absolute scale occur when the first bounce is also high. Some of these do slightly exceed the first bounce, but these cases are relatively few.
Visual inspection gives a general sense of the causes of the delayed rebound effect that is confirmed by systematic comparison. For the parameter ranges studied, 12.3% of combinations resulted in the delayed rebound effect. Of these , 93% occurred in cases where the first bounce was lower than the ICM prediction. However, having a first bounce lower than the ICM prediction was not highly predictive; only 22% of these cases have a higher second bounce.
Typically, the lower ball is not in contact with the floor when the balls collide for the second bounce, and the balls make a single contact with each other. When the lower ball is in contact with the floor, multiple contacts between the balls are possible, in a manner that is qualitatively similar to the first bounce. These cases lead to a noticeably lower second bounce than others in nearby parameter space. 
We show that the "delayed rebound effect," where the second bounce of two aligned balls is higher 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 is most prominent when the upper ball has a much smaller mass than the lower ball, and the distance gap between the balls is small when they are released. Typically, the finite duration of the collisions leads the first bounce to be smaller than predicted by the ICM, so the expected high bounce does not come until the second bounce. This is consistent with the informal observation that inspired this study: namely, that the delayed rebound effect is more commonly seen in ping-pong ball-basketball collisions than in tennis ball-basketball collisions.
Rather than focus on the parameters of a few particular sports balls, we look for universal behavior. The relevant ball parameters are reduced to the mass ratio µ = m 1 /m 2 and the coefficient of restitution ε, assumed to be the same for both balls. The initial conditions of drop height and gap between the balls is reduced to a single parameter τ d /τ f , which characterizes the time between the lower ball reaching the floor and the two balls making first contact. This approach represents a simplification over previous studies, and can be generalized to more than two balls.
We examined the details of the first bounce collisions, finding multiple contacts be- tween the balls in some cases. However, the multiple impacts did not correlate with the post-collision dynamics of the balls. For the first bounce, we found that small initial gaps resulted in bounces that were smaller than the ICM prediction, while the ICM is approximately correct for larger initial gaps, despite the overlapping collisions. This corresponds to experimental and theoretical results, albeit with a greatly simplified description of the contact force than in [16] . Direct comparison to realistic ball parameters for both first and second bounces is reserved for future work. While [9] suggests that the linear dashpot model gives qualitatively similar results to simulations using the Hertz force, [18] finds qualitatively different results between these forces when studying chain collisions. Thus, it may be worthwhile to repeat this study with a Hertzian contact force, more appropriate for viscoelastic spheres.
A thorough experimental study of the delayed rebound effect requires a mechanism to constrain the interacting particles to a single dimension. Precisely aligned spheres, as used in experimental studies of the first bounce, are unlikely to remain aligned for a second bounce. Low friction carts on an inclined track, with springs for repulsion, may be a useful model, although it might be difficult to achieve the range of mass ratios seen in ball drop experiments. expressions are explicitly real-valued, and match the expressions in the main text.
(20)
B Trajectory 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 damped harmonic oscillator X 1 (τ ) = 1 cos p cos τ 1 − ζ 2 1 + p e −ζ 1 τ − 1,
where τ = 0 is redefined as the time at which the lower ball first contacts the floor, and p = arctan
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 2 (0) = ∆H, the initial gap in dimensionless coordinates. The time-dependent position is
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 reason to prefer this method over the computational simulations performed. In either case, it is shown that the actual time time to collision is greater than τ d . Equation (22) assumes an adiabatic approximation, where the path of the lower ball is unaffected by collision(s) with the upper ball. The expression becomes exact in the limit µ → 0 or in the case τ d > τ f .