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People bouncing on trampolines: dramatic energy transfer, a table-top demonstration, complex dynamics and a zero sum game.

ABSTRACT: 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 game-theoretic 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.

SUBMITTER: Srinivasan M

PROVIDER: S-EPMC3827250 | biostudies-literature | 2013

REPOSITORIES: biostudies-literature

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People bouncing on trampolines: dramatic energy transfer, a table-top demonstration, complex dynamics and a zero sum game.

Srinivasan Manoj M Wang Yang Y Sheets Alison A

PloS one 20131113 11

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 o ...[more]

PMID: 24236029

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Project description:Research into the authenticity of facial emotion expressions often focuses on the physical properties of the face while paying little attention to the role of beliefs in emotion perception. Further, the literature most often investigates how people express a pre-determined emotion rather than what facial emotion expressions people strategically choose to express. To fill these gaps, this paper proposes a non-verbal zero-sum game - the Face X Game - to assess the role of contextual beliefs and strategic displays of facial emotion expression in interpersonal interactions. This new research paradigm was used in a series of three studies, where two participants are asked to play the role of the sender (individual expressing emotional information on his/her face) or the observer (individual interpreting the meaning of that expression). Study 1 examines the outcome of the game with reference to the sex of the pair, where senders won more frequently when the pair was comprised of at least one female. Study 2 examines the strategic display of facial emotion expressions. The outcome of the game was again contingent upon the sex of the pair. Among female pairs, senders won the game more frequently, replicating the pattern of results from study 1. We also demonstrate that senders who strategically express an emotion incongruent with the valence of the event (e.g., smile after seeing a negative event) are able to mislead observers, who tend to hold a congruent belief about the meaning of the emotion expression. If sending an incongruent signal helps to explain why female senders win more frequently, it logically follows that female observers were more prone to hold a congruent, and therefore inaccurate, belief. This prospect implies that while female senders are willing and/or capable of displaying fake smiles, paired-female observers are not taking this into account. Study 3 investigates the role of contextual factors by manipulating female observers' beliefs. When prompted to think in an incongruent manner, these observers significantly improve their performance in the game. These findings emphasize the role that contextual factors play in emotion perception-observers' beliefs do indeed affect their judgments of facial emotion expressions.

Dataset Information

People bouncing on trampolines: dramatic energy transfer, a table-top demonstration, complex dynamics and a zero sum game.

Publications

People bouncing on trampolines: dramatic energy transfer, a table-top demonstration, complex dynamics and a zero sum game.

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