{'fraction_non_alphanumeric': 0.046531892645521086, 'fraction_numerical': 0.015278261879865225, 'mean_word_length': 3.6389113446510373, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A unique pure mechanical system revealing dipole repulsion

We study multiple elastic collisions of a block and a ball against a rigid wall in one dimension. The complete trajectory of the block is solved as an analytic function of time. Near the turning point of the block the force carried by the ball is proportional to 1/x^3, where x is the distance between the wall and the block, in the limit that the block is sufficiently heavier than the ball. This is a unique pure mechanical system that reveals dipole-like repulsion.


I. INTRODUCTION
An elastic collision is a useful mechanical problem with which one can study energy, linearmomentum, and angular-momentum conservation in an explicit way. 1 While the final state is uniquely determined in a two-body elastic collision in one dimension, there are multiple solutions if three or more particles are involved. The reason is that the two constraints from energy and momentum conservation are not enough to determine the three or more final-state momenta. For example, a unique solution for a Newton's cradle of three or more pendula is obtained only if any two neighboring pendula are separated so that each collision involves only two pendula.
Various studies on such chain collisions have been carried out for the instantaneous contact force, 2-7 for the spring force, [8][9][10][11] and for the contact Hertz force. 7,8,10,12 In addition, there are numerous studies of the collisions of aligned balls in one dimension. [13][14][15][16][17][18][19][20][21] Harter investigated the velocity gain in such a chain collision. 16 Patricio studied the effect of the Hertz contact force in detail. 7 Redner considered the one-dimensional collisions of two cannonballs with a table-tennis ball sandwiched between them to derive a simple relation between that elastic collision and a corresponding billiard system. 22 In this paper, we consider the multiple collisions of two particles in one dimension against a rigid wall. As is stated above, this problem is, in principle, exactly solvable and we attempt to find the analytic solution to this problem. Many people, who have played or watched table-tennis games, are familiar with this multiple collision. When the server is waiting for the defender, the server often bounces the ball against the table repeatedly with the racket.
While the player presses down the ball with the racket, the speed of the ball keeps increasing and one can hear an interesting high-frequency sound near the turning point of the racket.
Whelan et al. studied a similar case under gravity focusing on its chaotic behavior. 23 Because the main feature of the problem stems from the velocity amplification of the light target against the heavy incident particle, we ignore any external forces other than the contact interaction between any two colliding objects. We construct a simple model system consisting of a block and a rigid wall that sandwich a ball. We derive the analytic expression for the complete trajectory of the block as a function of time. This new analytic solution reveals that, when the ball is sufficiently lighter than the block, the ball mediates a force proportional to 1/x 3 near the turning point of the block. Here, x is the distance between the wall and the block. This emergent dipole interaction is unique in a pure mechanical system.
In comparison with a previous study in Ref. 24 for a case with many balls, our derivation of the 1/x 3 rule is based on a complete analytic solution with a rigorous error analysis. This paper is organized as follows. In Sec. II we introduce our model, which simplifies the multiple collision problem of the table-tennis ball. In Sec. III we construct the simultaneous recurrence relation for the velocities of the colliding particles in each collision. Those for the position and time are also given. We determine the complete trajectory of the block as a function of time. In Sec. IV, we investigate the dynamics of the system in the continuum limit, where the block is sufficiently heavier than the ball and the block is near the turning point. Our conclusion is given in Sec. V, followed by the appendices, which contain useful mathematical formulas and some details of calculations.

II. THE MODEL SYSTEM
In this section, we define our model system. We restrict ourselves to elastic collision in one spatial dimension and ignore any external forces other than instantaneous contact forces.
respectively. After the first collision, the ball bounces against the wall and then they make the next collision. We call P n (t n , x n ) the n-th collision point between the ball and the block, where t n and x n are the time and position. The velocities of the block and the ball right after the n-th collision are U n and u n , respectively. The total number of collisions N is the smallest n that satisfies the condition U n ≥ |u n |. We also define the time interval ∆t n = t n+1 − t n between P n and P n+1 . Useful relations involving the computation of t n using ∆t n are given in Appendix A.
The elasticity of the collisions requires the conservation of energy as well as the linear momentum in the n-th collision: Note that the sign of u n is flipped in the first equation because the ball has bounced against the wall. With the initial conditions in Eq.
(2), one can solve U n and u n recursively. The first two pairs of the solutions are Based on Eqs. (4), we classify the ranges of the mass ratio α that determine the value of N .

III. ANALYTIC SOLUTIONS
In this section, we determine the point P n (t n , x n ) of the n-th collision between the ball and block, and the velocities U n and u n immediately after the collision.

A. Computation of U n and u n
We can reduce the recurrence relations for the velocities in Eq. (3) into the form where the 2 × 2 matrix A is defined by Applying Eq. (5) recursively, we determine U n and u n : Although our initial conditions are given in Eq. (2), the relation (7) is valid for any initial values of U 0 and u 0 .
The computation of U n and u n is rather tedious and we summarize the calculation in Appendix B. The results are where the parameter θ is related to the mass ratio α as Let us interpret the results in Eq. (8). Because the total kinetic energy is the sum of quadratic functions of U n and u n , the conservation of the total kinetic energy implies that U n and u n can be parametrized as a cosine and a sine of a common phase. The phase increases as a multiple of an elementary phase θ because the two eigenvalues of the non-Hermitian matrix A are a unimodulus complex number e iθ and its complex conjugate. According to these results, the block slows down to reach the turning point around nθ ≈ π 2 , where |u n | becomes the maximum. The maximum value for |u n | is larger than the initial speed U of the block with the enhancement factor 1/ √ α, if 0 < α < 1. From that moment to the region nθ ≈ π, the block bounces back and the ball loses the kinetic energy. If n = N , then the velocity −u n of the ball after the bounce against the wall must satisfy the constraint U n ≥ −u n .
Therefore, N must be the smallest integer that satisfies The recurrence relation of x n is derived in Eq. (A2) of Appendix A: By making use of Eqs. (8) and (9) and the elementary trigonometric identity sin α cos β ± cos α sin β = sin(α ± β), we find that With the initial condition x 1 = L, we find the analytic expression for x n as The analytic expression for x n is new. From this compact expression, it is easy to see that x n decreases to reach the minimum value x min near n ≈ 1 2 ( π θ − 1), where the denominator reaches its maximum. The turning point x min is reached at n = N 1 and the last collision is made at x N . C. Computation of N 1 and N From Eq. (13), we can determine that N 1 is where [x] + is the smallest integer that is greater than or equal to x. For a small α, θ ≈ 2 √ α In analogy to the derivation of Eq. (13), we can simplify U n + u n as where the final expression is valid for small α. This result is consistent with Eq. (6) of Ref. 22 with the uncertainty ±1. The quantity N max of that reference is approximately equal to 2N because, in that reference, the collision between the ball and the wall is also counted. In

D. Computation of t n
We have determined U n , u n , and x n for all n = 1, · · · , N . In this section, we use these results to compute the collision times t n . The computation of the time interval ∆t n is given in Appendix C. By making use of Eq. (A4), we sum the time intervals ∆t n in Eq. (C3) to find where τ is the time interval for the block to reach the wall when the ball is absent (α = 0): This analytic expression (17) for t n is new.
The complete trajectory of the block is where x n , U n , and t n are given in Eqs. (13), (8), and (17), respectively. In Fig. 3, we plot the trajectory x(t) of the block as a collection of straight line segments. The vertices represent the collision points P n (t n , x n ). The point on the vertical axis represents the initial collision point P 1 (0, L). The turning point is P N 1 (t min , x min ). When α = 0.25 (0.1), we have N 1 = 2 (3), and N = 3 (5). The x-t plot is convex-downward because every impact on the the block is along the positive x axis.
Next we find the turning point x min = x N 1 and corresponding time t min = t N 1 . According to Eq. (14), (N 1 − 1 2 )θ is bounded by Therefore, cos θ 2 ≤ sin(N 1 − 1 2 )θ ≤ 1. Substituting this constraint into Eq. (13) with n = N 1 , we can determine the range of x min as L sin θ With the values for sin θ 2 and tan θ 2 in Table I, we obtain When α is small, Therefore, considering both cases, we obtain where we used tan 2 1 2 θ = α. The errors of x min and t min are ±ατ , respectively.
It is interesting to notice that when α is sufficiently small, the time interval t min for the block to reach the turning point is approximately the same as τ , which is the time spent for the block to hit the wall when the ball is missing.

IV. CONTINUUM LIMIT
According to Eq. (C1), ∆t n becomes small when α is sufficiently small and n ≈ N 1 , where sin(n − 1 2 )θ ≈ 1. This does not require that α → 0 + . We call this the continuum limit. In this limit, both x n and t n can be treated to be continuous variables.
In this section, we investigate the dynamics of the system in the continuum limit. As the first step, we express the collision point x n in terms of t n . Next, we introduce a differentiable functionx(t) that may represent the trajectory of the block in the limit. By making use of the kinetic energy conservation, we interpret the kinetic energy of the ball as the potential energy of the block. Expressing the kinetic energy of the ball in terms of the differentiable functionx(t), we compute the force on the block in the continuum limit.

A. Relation between x n and t n
We observe that both x n in Eq. (13) and t n in Eq. (17) depend on n explicitly. In this section, we find the analytic expression for x n as a function of t n by eliminating the explicit n dependence.
From Eqs. (13) and (17), we have where sin θ 2 and tan θ 2 are replaced with the values listed in Table I. By making use of the fact that sin(n − 1 2 )θ ≥ 0 for all n ≤ N , we can express x n as a function of t n : where we used relations in Eq. (23) and Table I.
In Appendix D, we have shown that the complete trajectory x(t) of the block is always bounded by where the differentiable functions f (t) and g(t) are defined by As shown in Fig. 3, the lower bound f (t) passes every collision point P n and the upper bound g(t) is tangent to every line segment P n P n+1 .
In the continuum limit we can compute the velocity and the acceleration of the block The values forx,ẋ, andẍ must be understood as the time-averaged values. The uncertainties ofẋ(t) andẍ(t) can be obtained by replacing N with ∆.

B. Kinetic energy of the block
Based on Eqs. (29a) and (27a), we can determine the kinetic energy K of the block in terms of t or x as where K 0 = 1 2 M U 2 is the initial kinetic energy, which can be approximated as K 0 ≈ K(t = −∞). For a small α, the relative error of the kinetic energy is ± 1 2 α.

C. Potential energy of the block
We recall that the total kinetic energy is conserved during the whole process. Therefore, the sum of the kinetic energies of the block (K) and the ball (Φ) must be We interpret the kinetic energy Φ of the ball as the potential energy of the block. By making use of Eqs. (30a), (30b), and (31), we find that Next we compute the force, The force on the block has the same position dependence as an electric dipole interaction: In addition, x = x min when t = τ . Therefore, the duration of time to reach x min is the same as the case when the ball is absent. As is stated earlier, the derivation of the 1/x 3 rule originates from the analytic solution in Eq. (27), which is in contrast to the approach in Ref. 24 .
We can compute x min in the continuum limit. Setting Φ(x min ) = K 0 , we find that These values are consistent with Eqs. (21) and (22) within errors.

V. CONCLUSION
We have considered the one-dimensional elastic collisions of a ball and a block against a rigid wall. The initial state of the ball is u 0 = 0 at x = L and that of the block is U 0 = −U .
The trajectory x(t) of the block is completely determined in an analytic form. The analytic expression for the total number N of collisions between the block and the ball was also derived. The turning point of the block is ≈ L √ α at t ≈ τ = L/U , which is the time interval for the block to hit the wall when the ball is absent. Here α is the ratio of the mass of the ball to the mass of the block.
In the continuum limit where α is small and x(t) is near the turning point, the trajectory can be approximated as a differentiable functionx(t) in Eq. (27a). Because the total kinetic energy is conserved in this system, one can think of the kinetic energy of the ball as the potential energy of the block. Based on this idea, we have computed the force on the block and found that the force is proportional to 1/x 3 . It is remarkable that this is a unique pure mechanical system that reveals repulsive dipole-like interaction.
The time interval between P n and P n+1 is defined by ∆t n = t n+1 − t n which can be expressed in terms of U n or u n as where s n = x n + x n+1 is the distance that the ball travels between P n and P n+1 . Assuming u n < 0, we find that Substituting Eq. (11) into Eq. (A1), we can express ∆t n as The n-th collision time t n can be computed as Appendix B: Computation of U n and u n In this section, we solve Eq. (7), where the matrix A is defined in Eq. (6), We make a transformation diagonalizing A such that where R is a 2 × 2 matrix and λ i 's are the eigenvalues of the matrix A. Then, U n and u n are The i-th column vector R i of R satisfies the equation, for i = 1 and 2. The corresponding secular equation det[A − λ i 1] = 0, where 1 is the 2 × 2 identity matrix, is Solving this equation, we obtain where the parameter θ is related to the mass ratio α as In Sec. IV of Ref. 23  Here, we consider only the case 0 < α < 1. Note that θ = 0 for α = 0, 0 < θ < π 2 for 0 < α < 1, and θ = π 2 for α = 1. In the second and third columns of Table I, we list the values for trigonometric functions for 1 2 θ and θ, respectively. Now we determine R and R −1 as Substituting Eq. (B9) into Eq. (B4), we finally determine the velocities U n and u n as functions of n and α:  The time interval ∆t n = t n+1 − t n can be computed by substituting x n in Eq. (13) and U n + u n in Eq. (15) into Eq. (A3).
where τ is the time interval for the block to reach the wall when the ball is absent (α = 0), that is, τ = L/U . Substituting α = (n− 1 2 )θ and β = (n+ 1 2 )θ into the following trigonometric identity, we can simplify Eq. (C1) as Appendix D: Determination of the functions f (t) and g(t) The complete trajectory of the block x(t) is a set of line segments that connects consecutive collision point such that x(t n ) = x n , where We introduce a differentiable function, that passes every collision point P n and is convex downward. Therefore, at any time t ∈ where the equality holds only at t = t n for n = 1, · · · , N .
If there exists a curve g(t) that is tangent to every line segment P n P n+1 at t n ∈ [t n , t n+1 ] and convex downward, then g(t) is an upper bound of x(t) for all t ∈ [0, t N ]. We requirė and we set x n = g(t n ). According to Eq. (D4),ġ(t n ) =ḟ (t n+ 1 2 )/ cos θ 2 . Therefore, we find thatġ Note that t n ≤ t n ≤ t n+1 andġ(t n ) = −U cos n − 1 2 θ. Integratingġ(t) in Eq. (D6a) over t and imposing the boundary condition (D6c), we obtain g(t) = L (1 − t/τ ) 2 + α. (D7) As a result, the trajectory x(t) is bounded by for any t ∈ [0, t N ]. The left and right equalities hold at t = t n and at t = t n = t n+ 1 2 , respectively, for all n. In Fig. 3, we show the trajectories of the block x(t) at α = 0.25 and α = 0.1. As shown in this figure, the lower bound f (t) passes every collision point P n and the upper bound g(t) is tangent to every line segment P n P n+1 .
Tables 2 )θ can be obtained by using the identities such as cos(x ± δ) = cos x cos δ ∓ sin x sin δ and sin(x ± δ) = sin x cos δ ± cos x sin δ.