Figure 1.
A two-person game on a trampoline: Seat drop war.
Only one player shown. Each player alternatively bounces with her feet and her ‘seat’. The sequence of body configurations for one player is shown schematically. The other player goes through a similar sequence of configurations, but possibly with a phase difference.
Figure 2.
This figure shows a sequence of frames illustrating two balls dropped almost simultaneously onto a mini-trampoline, bouncing back up to very different heights. We see that the ball that makes contact second, rises much higher, as also seen in the mathematical models. See slow motion video in movie S2.
Figure 3.
Simple model of two people or two balls bouncing on a trampoline, as two point-masses on a massless trampoline.
a) The system can be in one of four phases: neither mass in contact with the trampoline (P0), only mass-1 in contact (P1), only mass-2 in contact (P2), and both masses in contact (P12). b) The geometry of the system is shown, along with the forces on the masses when both are in contact with the trampoline.
Figure 4.
a) The motion and
starting at rest from initial conditions
and
. b) The total energy (kinetic+gravitational potential) in mass-1 when both masses are in flight, as a fraction of the total energy. c) The state of the system when the right mass either just takes off (red dots) or when the right mass just lands (blue dots), when the left mass is already in the air.
Figure 5.
Energy transfer and contact time-lag.
Energy increase in mass-1 (red) and mass-2 (blue) as a function of the impact time-lag. The masses are equal and are placed symmetrically on the trampoline, so that . The separation between the masses
. a) The initial energies
are equivalent to dropping from rest from a height of 1 m. b) Mass-1's initial energy
is the same as for panel-a, but
. c) Mass-1's initial energy
is the same as for panel-a, except mass-2's energy is
.
Figure 6.
Basic mechanism of passive energy transfer from mass-1 to mass-2.
When mass-1 is moving up, the presence of mass-2 lowers the work done by the string on mass-1. Thus mass-1 takes off with lesser upward velocity than if mass-2 had not interfered.
Figure 7.
Stability of symmetric passive bouncing.
The four eigenvalues of the Jacobian (Floquet multipliers) corresponding to the mapping of the state over one period of the periodic motion: symmetric bouncing for symmetric masses. The product of the eigenvalues was equal to 1 (with an error of about ). Two eigenvalues are equal to
. In the intermediate regime shown, all four eigenvalues, two of which are complex conjugates and reciprocals of each other, have unit absolute values. At other regimes, one eigenvalue has magnitude greater than one, implying linear instability.
Figure 8.
a) Two players drop from the same height and they can pick a rigid leg length to modify their contact times. b) The normalized energy increase in mass-1 (the payoff function) as a function of the leg length choices of the two players, when the two players drop from an initial height of 2 m. The vertical line (blue/red) shows the minimax and maximin strategy that coincide. c) The payoff function when initial height is
0.9 m. For this height, the pure minimax (denoted player 2) and maximin (denoted player 1) strategy do not coincide. The second panel shows the optimal probabilities corresponding to mixed minimax strategies. For these calculations, we used parameter values pertaining to people, as used earlier (
kg, etc).
Figure 9.
Symmetric games: Dropping from the same height.
a) The normalized energy increase in mass-1 (the payoff function) as a function of the leg length choices of the two players, when the two players drop from an initial height of 2 m. The vertical line (blue/red) shows the minimax and maximin strategy that coincide. c) The payoff function when initial height is
0.9 m. For this height, the pure minimax (denoted player 2) and maximin (denoted player 1) strategy do not coincide. The second panel shows the optimal probabilities corresponding to mixed minimax strategies. For these calculations, we used parameter values pertaining to people, as used earlier (
kg, etc).
Figure 10.
All parameters are kept at their default symmetric values, except for the one parameter that is varied for each panel. a) Players drop from different initial heights. Player 1 has initial height m; player 2's initial height is varied as shown. All else is symmetric. The player that starts higher, that is, has higher initial energy, gains even more energy in a single bounce, further increasing her energy. b) Players have different masses. Player 1's mass
is kept at 70 kg and player 2's mass
is varied as shown. Again, the player with greater initial energy gains even more energy. c) Players have different initial speeds. Player 1 starts at rest and player 2's initial speed
is changed. When player 2 has a positive or upward initial speed, it lands second and gains energy from player 1. d) Players have asymmetric positions on the trampoline: player 2's position is moved to the right by
. We find that the player that is closest to its end of the trampoline loses more energy.
Figure 11.
Statistical mechanics of many balls bouncing passively.
a) Twenty five balls are dropped from approximately but not exactly the same height. b) The angle of the string is shown as a function of time. Note that the string oscillates macroscopically initially, as the masses bounce together coherently on it. However, eventually this macroscopic coherent motion of the masses and the coherent oscillatory string motion gets ‘damped out’ with the energy getting transferred to incoherent motion of the masses. c) A ‘macroscopic’ kinetic energy computed as with mean ball velocity
decreases with time, as the masses' velocities cancel each other more.