1. Observation: Observe the billiard ball model of colliding elastic circles in two dimensions, representing perfectly elastic billiard balls on a perfectly elastic table. Watch the red ball for awhile and try to estimate the average distance it travels between collisions with other balls. Then, using the Pause key on the computer, freeze the screen at end appropriate time and measure the lengths of the tracks. Measure these lengths from the screen and record them. Continue until you have measured 30 lengths.
Analysis: Group the lengths into a frequency distribution, recording the number of lengths in each of the following ranges: >= 0 and < 4 cm, >=4 cm and < 8 cm, >= 8 cm and < 12 cm, etc., up to the highest range you have observed.
Construct a bar graph showing this frequency distribution.
Sketch a smooth curve which you think represents the actual frequency distribution for these distances. Imagine that you had been able to observe trillions of lengths and that you had put them into very small ranges, like 0 to .01 cm, .01 to 2 cm, etc.. The curve you draw will represent the tops of the bars on your bar graph.
2. Observations: Observe the billiard ball model of colliding elastic circles in two dimensions. Get a feel for how the velocities of the red ball are distributed. If the observed range of velocities is partitioned into five equal intervals, what percent of the velocities would go into each? Sketch a bar graph indicating your estimates.
Analysis: Now use the pause key to randomly stop the simulation, using a delay of at least three seconds between stopping times and without looking at the screen during that delay (this is in order to avoid any prejudice in your timing of the pause). Read the velocity of the red ball from the screen. Obtain at least 50 velocities in this manner.
Group your velocities into five equal intervals and determine the number of velocities, and the percent of all the observed velocities, which fall into each interval. Sketch a bar graph of your results. Compare with the graph you sketched from your previous estimates.
3. Observations: Observe the billiard ball model of colliding elastic circles in two dimensions. Observe the total x and y energies to see how they vary. Decide whether you think that the x and y energies would tend to be equal, on the average.
Prediction: Focus on just the x energy. Make a sketch depicting how you think the x energies might be distributed around the average energy (e.g., if you divided the range of energies into a large number of intervals, what might the history graham look like?).
Observation: Now take some data for the x and y energies. Every couple of seconds, press the Pause key on your computer and record the x and y energies. Do this until you have read about 30 sets of x and y energies.
Prediction and Analysis: Would you expect one or the other to be greater? Estimate the average x energy and the average y energy. Does it look like one average is greater than the other?
Put your information into Excel or into you calculator and determine the averages of the x and y energies.
Do you think the difference in the averages is significant or just due to statistical uncertainty?
4. Observation: Run a neural net simulation and sample 'best-so-far' and 'recent average' for trial numbers 50, 100, 200, 300, 400, 500, 750 and 1000.
Analysis: Sketch a graph showing both results vs. trial number.