# Bouncing Balls

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30-Dec-2015Category

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### Transcript of Bouncing Balls

Bouncing BallsAlex Jing Wei Huang

Goals of this ProjectAnalyze the ball collision rates under different conditions (parameters). Different container shapes). Different numbers of balls, Different initial velocity ranges.Analyze the velocity distribution of these balls.

State Space of the SystemState space: 4 dimension.Position (x and y coordinates)Velocity (x and y directions)

Other parameters of the systemShape of the containerRadius of balls

Simulation ToolsPythonPygame pakageVisual pakage

Properties of the SystemBalls are moving in 2D space.Balls have random initial velocities.All balls have same radius and mass.Balls can collide with walls and other balls.Energy is conserved during all time.

Experiment 1: The collision rates with respect to different container shapes10 balls (with random initial velocities between 5 to 5) are put in three containers with same area and different shape.600 x 600 (pixel x pixel)400 x 900300 x 1200We calculate the number of collisions after moving for a day (24*602 seconds).Note by second, I mean iteration.

Experiment 1 (cont.)

Experiment 1 (cont.)600 x 600 square window

Data 1 Data 2Data 3 Data 4Data 5AverageBall to wall837385657114747157557455,6Ball to ball967369057186535743876721.6Total14177.2

Experiment 1 (cont.)900 x 400 rectangular window

Data 1 Data 2Data 3 Data 4Data 5AverageBall to wall903279037539944269298169Ball to ball654351658268676844576240.2Total14409.2

Experiment 1 (cont.)1200 x 1200 rectangular window

Data 1 Data 2Data 3 Data 4Data 5AverageBall to wall1040910688932396961040010103.2Ball to ball584062054710626459145786.6Total15889.8

Experiment 1 (cont.)Summary of data

600 x 600900 x 4001200 x 300Ball to wall7455,6816910103.2Ball to ball6721.66240.25786.6Total14177.214409.215889.8

Experiment 1 (cont.)ConclusionWe can minimize the number of ball-to-wall collisions by putting them in a square container.

Experiment 2: Collision rates with respect to different numbers of ballsWindow size: 600 x 600 .Number of balls101520

Experiment 2 (cont.)10 balls (same data from Experiment 1)

Data 1 Data 2Data 3 Data 4Data 5AverageBall to wall837385657114747157557455,6Ball to ball967369057186535743876721.6Total14177.2

Experiment 2 (cont.)15 balls

Data 1 Data 2Data 3 Data 4Data 5AverageBall to wall107091118812681118761175311641.4Ball to ball130041375614361139291383413776.8Total25418.2

Experiment 2 (cont.)20 balls

Data 1 Data 2Data 3 Data 4Data 5AverageBall to wall150231760015865180951320915958.4Ball to ball226272909025182291672214025641.2Total41599.6

Experiment 2 (cont.)Summary of data

ConclusionBoth ball-to-wall and ball-to-ball collisions increase as the number of balls increases.

10 balls15 balls20 ballsBall to wall7455,611641.415958.4Ball to ball6721.613776.825641.2Total14177.225418.241599.6

Experiment 3: Collision rates with respect to different initial velocity rangesWindow size: 600 x 600 .Number of balls : 10Initial velocity ranges[ -5, 5] in each x and y direction[-10, 10] [-15, 15]

Experiment 3 (cont.)Initial velocity range : [-5, 5]

Data 1 Data 2Data 3 Data 4Data 5AverageBall to wall837385657114747157557455,6Ball to ball967369057186535743876721.6Total14177.2

Experiment 3 (cont.)Initial velocity range : [-10, 10]

Data 1 Data 2Data 3 Data 4Data 5AverageBall to wall125931380113514142091685914195.2Ball to ball114181117610218118091287811499.8Total25695.0

Experiment 3 (cont.)Initial velocity range : [-15, 15]

Data 1 Data 2Data 3 Data 4Data 5AverageBall to wall177012376621139173122228120439.8Ball to ball133011717314962138301604815062.8Total35502.6

Experiment 3 (cont.)Summary of data

ConclusionBoth ball-to-wall and ball-to-ball collisions increase as velocity range increases.

[-5, 5][-10, 10][-15, 15]Ball to wall7455,614195.220439.8Ball to ball6721.611499.815062.8Total14177.225695.035502.6

Velocity DistributionBalls velocities are changing during the experiment (due to collisions with each other). We analyze the velocity change by plotting the histogram of these balls velocities at each time step.

Velocity Distribution (cont.)The simulation verifies that, after certain period of time, the balls velocities will follow the Boltzmann distribution.

For proof, please look up Wikipedia

Future ExplorationHow the collision rate is affected if the balls are moving in 3D?if the container is a triangle or a circle?if balls have different radius and mass?

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