1. Bouncing Balls Alex Jing Wei Huang

2. Goals of this Project Analyze 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.

3. State Space of the System State space: 4 dimension. –Position (x and y coordinates) –Velocity (x and y directions) Other parameters of the system –Shape of the container –Radius of balls

4. Simulation Tools Python –Pygame pakage –Visual pakage

5. Properties of the System Balls 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.

6. Experiment 1: The collision rates with respect to different container shapes 10 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 900 –300 x 1200 We calculate the number of collisions after moving for a day (24*60 2 seconds). –Note by “second”, I mean iteration.

7. Experiment 1 (cont.)

8. 600 x 600 square window Data 1Data 2Data 3Data 4Data 5Average Ball to wall837385657114747157557455,6 Ball to ball967369057186535743876721.6 Total14177.2

9. Experiment 1 (cont.) 900 x 400 rectangular window Data 1Data 2Data 3Data 4Data 5Average Ball to wall903279037539944269298169 Ball to ball654351658268676844576240.2 Total14409.2

10. Experiment 1 (cont.) 1200 x 1200 rectangular window Data 1Data 2Data 3Data 4Data 5Average Ball to wall1040910688932396961040010103.2 Ball to ball584062054710626459145786.6 Total15889.8

11. Experiment 1 (cont.) Summary of data 600 x 600900 x 4001200 x 300 Ball to wall7455,6816910103.2 Ball to ball6721.66240.25786.6 Total14177.214409.215889.8

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

13. Experiment 2: Collision rates with respect to different numbers of balls Window size: 600 x 600. Number of balls –10 –15 –20

14. Experiment 2 (cont.) 10 balls (same data from Experiment 1) Data 1Data 2Data 3Data 4Data 5Average Ball to wall837385657114747157557455,6 Ball to ball967369057186535743876721.6 Total14177.2

15. Experiment 2 (cont.) 15 balls Data 1Data 2Data 3Data 4Data 5Average Ball to wall107091118812681118761175311641.4 Ball to ball130041375614361139291383413776.8 Total25418.2

16. Experiment 2 (cont.) 20 balls Data 1Data 2Data 3Data 4Data 5Average Ball to wall150231760015865180951320915958.4 Ball to ball226272909025182291672214025641.2 Total41599.6

17. Experiment 2 (cont.) Summary of data Conclusion –Both ball-to-wall and ball-to-ball collisions increase as the number of balls increases. 10 balls15 balls20 balls Ball to wall7455,611641.415958.4 Ball to ball6721.613776.825641.2 Total14177.225418.241599.6

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

19. Experiment 3 (cont.) Initial velocity range : [-5, 5] Data 1Data 2Data 3Data 4Data 5Average Ball to wall837385657114747157557455,6 Ball to ball967369057186535743876721.6 Total14177.2

20. Experiment 3 (cont.) Initial velocity range : [-10, 10] Data 1Data 2Data 3Data 4Data 5Average Ball to wall125931380113514142091685914195.2 Ball to ball114181117610218118091287811499.8 Total25695.0

21. Experiment 3 (cont.) Initial velocity range : [-15, 15] Data 1Data 2Data 3Data 4Data 5Average Ball to wall177012376621139173122228120439.8 Ball to ball133011717314962138301604815062.8 Total35502.6

22. Experiment 3 (cont.) Summary of data Conclusion –Both ball-to-wall and ball-to-ball collisions increase as velocity range increases. [-5, 5][-10, 10][-15, 15] Ball to wall7455,614195.220439.8 Ball to ball6721.611499.815062.8 Total14177.225695.035502.6

23. Velocity Distribution Balls’ 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.

24. 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

25. Future Exploration How 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?