Preliminary Observation Run the program billiard simulation. Simply open the simulation and hit the 'Enter' key. Watch the KEx and KEy values as they change with each collision, representing the total x and y kinetic energies of the particles. One of the green particles traces out a path as it moves across the screen. This is the particle whose speed is indicated next to the word 'speed' (about halfway down the window, toward the right-hand side). Most of the time when this particle collides with another its speed changes. Watch for a minute or so and see if you can learn to estimate its speed before looking at the posted speed. How long does it take to move a distance equal to the height or width of the screen when its speed is 10? How long should it then take to move the same distance if its speed is 5? Is that about what you observe? **** It takes about 2.5 seconds for it to make it across the screen with a speed of 10 and about 5 seconds when the speed is 5. It should take about double the time to move the same distance when the speed is 5. #$&* How frequently does that green particle collide with other particles? What percent of the time intervals between collisions do you think are less than a second? What percent are less than 2 seconds? What percent are less than 4 seconds? What percent are less than 10 seconds? **** I would say about 5 percent of the time the intervals are less than a second, about 15% of the time the intervals are less than 2 seconds, about 60% of the time they're less than 4 seonds, and about 90% of the time they are less than ten seconds. #$&* Watch the 'red' particle for a couple of minutes, estimating the average time between its collisions and its average speed. What percent of the time intervals between collisions do you think are less than a second? What percent are less than 2 seconds? What percent are less than 4 seconds? What percent are less than 10 seconds? At its average speed, how long do you think it would take to move a distance equal to the height or width of the screen? On the same scale you used for the speed of the green particle, what do you think is the average speed of the red particle? **** I think it would take about 12 seconds to get across the screen. It's average speed is about 3 or 4. I would say about 1 percent of the time the intervals are less than a second, about 3% of the time the intervals are less than 2 seconds, about 10% of the time they're less than 4 seonds, and about 30% of the time they are less than ten seconds. #$&* Watch the 'blue' particle, and speculate on what property of this particle is different from that of the other particles. The blue particle could be heavier than the other green ones, causing it to speed up less when hit. ---------------------------------------- Experiment kinmodel_01: The Distribution of Atomic Speeds When the speed of the simulation is moderate it is possible to watch a specific particle (the red particle or the blue particle in the default simulation) and obtain an intuitive feeling for the relative frequencies of various speeds. Run the simulation billiard simulation at the default settings. Observe the simulation long enough to get a feel for the maximum velocity you are likely to see. Then estimate how much time it spends at slow (less than 1/3 of max vel.), medium (between 1/3 and 2/3 of max. vel.) and fast (more than 2/3 of max. vel.) velocities. Express your estimates in percents of the total time spent in the three different velocity ranges. **** The max speed is probably about 15. I would say it spends about 25% of it's time below 1/3 max velocity, about 60% between 1/3 and 2/3 of max, and 15% above 2/3 of max velocity. #$&* Draw a histogram (a bar graph) of your estimates. Describe your histogram. **** The histogram has three bars, and the middle bar (between 1/3 and 2/3 max), is the tallest. Then the bar to the left (less than 1/3 max) is the second tallest. The right bar (2/3 max and greater) is the smallest. #$&* Now suppose you had estimated the percent of time spent in each of 10 velocity ranges (i.e., from 0 to .1 of max. vel., .1 to .2 of max. vel., etc, up to max. vel.). From your previous estimates, without further viewing the simulation, make a reasonably consistent estimate of the proportion of time spent in each of these ranges. Sketch a histogram of your estimates and describe the graph in your writeup. **** The histogram goes from 0 to .1, .1 to .2, .2 to .3, .3 to .4, .4 to .5, .5 to .6, .6 to .7, .7 to .8, .8 to .9, and .9 to 10. The bars, starting from left to right, go up to the percentage marks of: 5%, 7%, 9%, 20%, 25%, 11%, 8%, 7%, 5%, 3%. #$&* Sketch the smooth curve you think best represents the distribution, with the curve being highest at the most likely speed, near the horizontal axis for speeds you very seldom observe. According to your sketch, which speed is the most likely? What percent of the area under your curve corresponds to speeds within one unit of your most likely speed (e.g., if your most likely speed was 3, you would estimate the area under the curve between speed 3 - 1 = 2 and speed 3 + 1 = 4). For what speed(s) is the curve half as high as the maximum? For what speed(s) is it half this high? **** The most likely speed is about 7. About 56% of the speeds is under this part of the curve. The speeds that the curve is half as high are 4.5 and 9. It's half that high again at about 2 #$&* Watch the green particle for long enough to estimate the percent of time it spends at speeds more than 2 units greater than the most likely speed, but not more than 4 units greater. What percent of the time do you estimate that the green particle is moving at less than half its most likely speed? **** Around 25% #$&* Watch the number corresponding to the speed of the green particle. Close your eyes for a few seconds at a time and open them suddenly, and each time write down the velocity of the particle as you see it immediately after your eyes open. Record about 100 velocities in this manner. **** I recorded the times in a chart, putting a tally mark each time I saw that number appear, so the totals will show my numbers: 1: 3 times 2: 6 times 3: 5 times 4: 11 times 5: 13 times 6: 21 times 7: 14 times 8: 12 times 9: 9 times 10: 3 times 11: 1 time 12: 1 time 14: 1 time #$&* Tally your velocities to see how many of the 100 velocities were 0, how many were 1, how many were 2, etc. Construct a histogram of your results and compare to the histograms you predicted earlier. **** The results were close to my histogram earlier. Instead of 7 being the most common speed, it was six. Then it descended on both sides, however the left side was slightly higher than the right end side (excluding the middle), which slightly disagreed with my other histogram. #$&* Experiment kinmodel_02: Mean free path; mean time between collisions It is possible to observe the mean free path of the green particle between collisions. First observe the particle for a few minutes and try to get a feel for how the distances traveled between collisions with other particles are distributed. Make your best estimate of what percent of the time the particle travels less than 1 inch between collisions, the percent of the time the distance is between 1 and 2 inches, the percent of the time the distance is between 2 and 3 inches, etc.. When the particle collides with a 'wall', it doesn't count as a collision and distance keeps accumulating until it collides with another particle. Sketch a histogram of your estimates, and also document the distance on your monitor between the 'walls' that confine the particles. **** The walls are about 6 in on my screen. 0-1 in:10% 1-2 in:15% 2-3 in:25% 3-4 in:30% 4-5 in:15% 5-6 in:5% This histogram looks like a bell curve and the max is in the region of 3-4 inches. #$&* Now take some data. Using the 'pause' and 'restart' buttons, stop and start the particle motion as required in order to observe the distances traveled by the green particle between collisions. Create a ruler using a strip of paper whose length is equal to the diagonal of the 'box' within which the particles move. Mark the strip into 16 equal segments (you can easily do this by folding the strip in half, lengthwise, four times in succession, then numbering the folds from 1 to 15). Use this ruler to measure distances traveled. Don't leave any distances out, because this would bias the sample. Observe at least 30 distances. Describe how you obtained your data and report your data as a frequency distribution (i.e., the number of observations for which the distance rounded to 0, 1, 2, 3, ..., inches). **** I took a sheet of paper and marked it at sixteen intervals then measured the distance travelled by pausing and unpausing at collisions. 0 in:3 1 in:4 2 in:4 3 in:6 4 in:8 5 in:3 6 in:2 #$&* Sketch a histogram of your results. Sketch the histogram you would expect from a large number of observations. Describe your histograms, and how they compare with your previous predictions. **** The histogram is fairly similar to the one earlier. The highest bar is 3-4 and the left end is higher than the right end (excluding middle). #$&* ALTERNATIVE Start the program using default values. Let it run for several seconds, then start observing the green particle. Keep track of whether it is moving more in the x or more in the y direction. Just say to yourself 'x x x y y y y y x x y x y y y ... ', according to what you see. Do this at a steady but comfortable pace. Continue this for a minute or so. Then take a pencil and paper, or alternatively open a text editor in a separate window, and start writing down or typing your x and y observations. I just did this and in about a minute or two I got the following: xxyyyyxyyxxyxyyxxyxxxyyyxxyyxxyyxyxxyyyxyyyxyyxy. I haven't done this before and found this a little confusing. Every time the particle got hit I wanted to type a letter right away, but I hadn't had time to figure out in what direction it was headed. With practice I began to get over that. You will experience different glitches in the process, but with a few minutes of practice you'll be able to do a reasonably good job. I suspect I also had some tendency to type one of the letters in preference to the other (e.g., x in preference to y, or maybe y in preference to x). I don't recommend fighting this sort of tendency but just noticing it and gently trying to improve. I didn't do this with pencil and paper, and it would be interesting to see if the tendencies are the same when writing as opposed to typing. However that's not our purpose here. As an alternative, you could make marks on a piece of paper then type them out (you might even use simple vertical and horizontal dashes, like | and -, which you can then translate into y's and x's). At whatever pace you prefer, write or type about 50 observations of x or y. List them here. **** yxyxyyxxyxyxyxxxxyyxxxxyxyyxxyyxyxyxxxyxxxyyxxyxyx #$&* Now notice the KEx and KEy values represented toward the right-hand part of the program's window, just a little ways below the middle of the screen. KEx represents the total x component of the kinetic energies of all the particles and KEy the total y component. Using the Pause and Restart buttons, stop and start the program and with each stop record the KEx and KEy. Values can be rounded to the nearest whole number. After each observation quickly hit 'Restart' then 'Pause', and record another. Record about 50 observations. Having recorded the 50 KEx and KEy values, write 'x' next to each pair for which the x value is greater, 'y' next to each pair for which the y value is greater. List your x's and y's in sequence here (don't list your values for the KE). **** yyxyxyyxxyxxxxxxyyyxyyyxxxyxxyyxyxxxyyxyxxyxyxxxyx #$&* What is the greatest KEx value you observed and what is the least? **** 1905 and 1321 #$&* What is the greatest KEy value you observed and what is the least? **** 1898, 1263 #$&* On a 50-trial sample of a normal distribution, the mean would be expected to occur about halfway between the least and greatest values observed, and the expected standard deviation would be very roughly 1/5 of the difference between the least and greatest values. According to this (very approximate) rule, what would be the mean and standard deviation of your KEx values, and what would be the mean and standard deviation of your KEy values? **** KEx mean:1613 s.d.: 116.8 KEy mean:1581 s.d.: 127 #$&* Do you think the mean KEx value differs significantly from the mean KEy value? There is a difference. By 'significantly', we mean a difference that seems greater than what would naturally occur by chance statistical variations. **** The KEx is higher than the KEy value, and differs less. #$&* Experiment kinmodel_04: The improbability of all particles being segregated on one side of the viewing area (order vs. disorder) Any selected region of the screen can be selected for viewing by masking the rest of the screen. The viewer can estimate the probability of this region being vacated within an hour, within a day, within a year, ..., within the age of the universe. Results will differ with the size of the region, the number of particles and the speed of the simulation. Cut out a 1-inch square and watch the simulation for 2 minutes on the middle default speed. Observe how many times the square becomes 'empty' of particles. Estimate what percent of the time this square is empty. **** It was empty about 15% of the two minutes. #$&* Enlarge the square to a 1-inch by 2-inch rectangle and repeat. **** It was empty about 5% of the time. #$&* Enlarge to a 2-inch by 2-inch square and repeat. **** It was empty only for a second. #$&* Enlarge this square to a 2-inch by 4-inch rectangle and repeat. **** It was not empty the entire time. #$&* Enlarge to a 4-inch by 4-inch square and repeat. **** It was not empty the entire time. #$&* Mask all but 1/4 of the screen and repeat. **** It was not empty the entire time. #$&* How long do you think it would take, on the average, for 1/4 of the screen to become completely empty of particles? **** It would time maybe an hour or a couple hours for chance to make that emprty. The box on my screen is about 3x3 inches and there's usually a minimum of 2 balls in the box. #$&* How long do you think it would take, on the average, for 1/2 of the screen to become completely empty of particles? **** I don't think that would happen. If it would, it would take a long long time. #$&* A typical closet is about 100 million times as far across as the distance represented by the screen. Ignoring for the moment that the closet is three-dimensional and hence contains many more air molecules than would be represented by a 2-dimensional simulation, how long do you think you would have to wait for all the molecules to move to one side of the closet? **** I don't think that will ever happen because of diffusion properties.