People Bouncing on Trampolines: Dramatic Energy Transfer, a Table-Top Demonstration, Complex Dynamics and a Zero Sum Game

Complex Dynamics and a Zero Sum Game. PLoS ONE 8(11): e78645. doi:10.1371/journal.pone.0078645 People Bouncing on Trampolines: Dramatic Energy Transfer, a Table-Top Demonstration, Complex Dynamics and a Zero Sum Game Manoj Srinivasan 0 Yang Wang 0 Alison Sheets 0 Assad Anshuman Oberai, Rensselaer Polytechnic Institute, United States of America 0 Mechanical and Aerospace Engineering, The Ohio State University , Columbus, Ohio , United States of America Jumping on trampolines is a popular backyard recreation. In some trampoline games (e.g., ''seat drop war''), when two people land on the trampoline with only a small time-lag, one person bounces much higher than the other, as if energy has been transferred from one to the other. First, we illustrate this energy-transfer in a table-top demonstration, consisting of two balls dropped onto a mini-trampoline, landing almost simultaneously, sometimes resulting in one ball bouncing much higher than the other. Next, using a simple mathematical model of two masses bouncing passively on a massless trampoline with no dissipation, we show that with specific landing conditions, it is possible to transfer all the kinetic energy of one mass to the other through the trampoline - in a single bounce. For human-like parameters, starting with equal energy, the energy transfer is maximal when one person lands approximately when the other is at the bottom of her bounce. The energy transfer persists even for very stiff surfaces. The energy-conservative mathematical model exhibits complex non-periodic long-term motions. To complement this passive bouncing model, we also performed a gametheoretic analysis, appropriate when both players are acting strategically to steal the other player's energy. We consider a zero-sum game in which each player's goal is to gain the other player's kinetic energy during a single bounce, by extending her leg during flight. For high initial energy and a symmetric situation, the best strategy for both subjects (minimax strategy and Nash equilibrium) is to use the shortest available leg length and not extend their legs. On the other hand, an asymmetry in initial heights allows the player with more energy to gain even more energy in the next bounce. Thus synchronous bouncing unstable is unstable both for passive bouncing and when leg lengths are controlled as in game-theoretic equilibria. - Bouncing on a trampoline has evolved from a backyard activity for children to an Olympic Sport. While Olympic trampolining only has one person bouncing on a trampoline, in its recreational form, it is quite common for more than one person to bounce on the trampoline simultaneously. In particular, children play a twoperson game on trampolines called seat drop war. In this game, each player bounces alternatively with her feet and her seat (being in an L-shaped body configuration), as shown in Figure 1. See also movie S1, showing this game being played. Each player is able to increase her mechanical energy while bouncing (jumping) with her feet by performing mechanical work with her legs, but she essentially bounces passively when bouncing on her seat (and probably loses some energy due to damping). The goal of this game is to be the last player bouncing. As the game progresses with the two players bouncing alternatively with their feet and their seat, the relative phasing between their bounces typically changes: sometimes the players bounce out of phase and sometimes they bounce in phase. The game often ends with one person having so little upward velocity when bouncing on her seat that she is unable to get back on her feet for the next bounce. Often, associated with this loss, the second player appears to have gained most of the energy lost by the first player, thereby bouncing higher than usual. This article is motivated by this apparently dramatic energy transfer between the players, which typically happens during a bounce in which the two players are simultaneously in contact with trampoline for some overlapping time period. Here, we show that the dramatic energy transfer is observed even in the passive bouncing of inanimate masses. We first describe a simple physical demonstration of the energy transfer: dropping two balls simultaneously onto a small trampoline sometimes results in one ball bouncing much higher than the other. Then, we construct a simple energy-conservative mathematical model, with the two people modeled as masses bouncing passively on a trampoline. This model also exhibits the dramatic energy transfer observed in seat drop war. We call the energy transfer dramatic because essentially all the energy of one person/ball gets transferred to the other in a single brief interaction. We show that there is typically an optimal difference between the landing times of the two masses (hereafter called the contact time-lag) that maximizes energy transfer. The mathematical model, in absence of dissipation or sideways movement of masses, displays complex non-periodic motion, with repeated transfer of energy between the two masses. Finally, we make a first step at analyzing the game, not as a simple passive dynamics problem involving two balls bouncing, but as a strategic competitive game between two players from a game theoretic perspective, obtaining the optimal strategies for the two players for the zero-sum game. A physical demonstration: Two balls on a trampoline To illustrate that energy transfer between people on a trampoline can happen through purely passive mechanisms, we designed a simple table-top demonstration involving a store-bought minitrampoline and two balls (see also Materials and Methods). Figure 2 shows a series of key frames, illustrating the energy transfer between the two (tennis) balls, dropped nearly but not exactly simultaneously. The two balls contact the trampoline at slightly different times, with some overlapping period when they are both in contact with the trampoline. The mass that makes contact with the trampoline second bounces much higher. See also movie S2, which shows this specific example in slow motion, and also other examples illustrating how when the masses make contact with the trampoline approximately simultaneously, they bounce up to about the same height. We did not perform carefully controlled drops, make detailed measurements of the resulting bounces, or try to make this tabletop demonstration a dynamically scaled version of two humans bouncing on a larger trampoline. We intend this only as a demonstration of the phenomenon. When dropped by human hands, the two balls often land at slightly different times due to human motor variability, resulting in different amounts of overlap between their contact phases with the trampoline. As a consequence, as seen from the mathematical models below, the rise heights of the masses after the bounce have corresponding variability. When there is no contact overlap, as happens often (if the drops are not nearly simultaneous), there is (...truncated)