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course Phy 122
IntroductionThe program kinmodel_.EXE simulates in 2 dimensions the kinetic behavior of a user-specified number of spherical atoms with user-specified masses, colliding as hard elastic disks at a user-specified center-to-center distance. The initial positions and speeds of the particles are randomly generated by the computer and the simulation develops from the corresonding initial state. Information related to particle speeds, x- and y- kinetic energies, and energy distributions is provided in the form of unlabelled graphs on the screen.
The simulation can be stopped once it is running by striking the 's' key.
The default settings
The default settings are chosen to provide 43 particles or atoms, 32 colored green, 8 colored dark blue, 2 colored light blut and one colored red. One of the 'green' particles starts at rest and leaves a trail when it moves.
estimating the distribution of atomic speeds and mean free path
equipartition of energy and the tendency for an ordered system to move toward disorder
the improbability of 43 particles being segregated on one side of the viewing area (unlikelihood of an ordered configuration)
the probability that a particle's speed will occur in a given range
the connection between relative particle mass and average speed
the development of empirical frequency vs. speed and frequency vs. energy histograms (order and disorder, this time with statistical order emerging from the disordered system)
images of 2-dimensional collisions
appreciation of time scale of kinetic interactions in a gas at typical pressures and temperatures (at medium default speed the simulation represents many of the features of a thin slice approximately 10 nanometers on a side and, say, a nanometer thick, of a monatomic gas at room temperature and several atmospheres pressure, with 1 second of real-world time corresponding to a few thousand years of simulation time).
and others.
Experiments and Activities
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?
When its speed is 10, its takes about 2 seconds to move across the screen. It takes twice as long when its speed is 5.
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• 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 believe it collides with a particle about every second. I would say 50% less than a second. 75% every 2 seconds. 90% every 4 seconds. 100% less than 10 seconds.
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• 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 would say about 20% less than a second. 40% less than 2 seconds. 60% less than 4 seconds and 90% less than 10 seconds. The average speed of the red particle is about a 1.
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• Watch the 'blue' particle, and speculate on what property of this particle is different from that of the other particles.
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.
Maximum speed is about 15 - 10% of time
Slow - 60 % of the time.
Medium - 30 % of the time.
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• Draw a histogram (a bar graph) of your estimates. Describe your histogram.
The bar graph I have drawn shows that a majority of the time the average speed is in the slower range, less than 5.
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• 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 bar graph would show that a peak is the slower time of 0.3 to 0.4 max velocity and taper off in each direction from there.
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• 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 between four and 5. That would be the peak and then the bell curve would taper down in both directions from there.
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• 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?
I would estimate that it spends about 20% at speed greater than 2 units above the average.
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• 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.
5, 2, 2, 2, 2, 6, 8, 7, 4, 8, 7, 4, 5, 6, 3, 5, 7, 6, 5, 13, 1, 6, 7, 7, 7, 4, 5, 9, 8, 4, 6, 9, 5, 11, 8, 3, 9, 4, 6, 3, 2, 6, 7, 14, 6, 3, 7, 7, 5, 5, 7, 8, 3, 1, 2, 2, 5, 3, 5, 6, 5, 6, 11, 12, 9, 6, 4, 7, 8, 7, 10, 6, 2, 0, 5, 6, 4, 8, 3, 5, 4, 3, 5, 5, 3, 2, 1, 11, 12, 7, 0,
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• 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.
0 -2
1 - 3
2 - 9
3 - 8
4 - 8
5 - 13
6 - 13
7 - 13
8 - 7
9 - 4
10 - 1
11 - 3
12 - 2
13 - 0
14 - 1
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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.
Less than 1 inch - 10%
1 to 2 inches - 35%
2 to 3 inches - 25%
3 to 4 inches - 20%
Greater than 4 inches - 10%
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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 measured the screen then divided the paper into 16 different squares. I then observed 30 different distnaces.
2, 10, 1, 8, 1, 2, 1, 1, 13, 2, 2, 3, 5, 1, 8, 2, 3, 3, 5, 9, 3, 1 , 10, 2, 5, 5, 6, 3, 1, 2,
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• 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 shows that a majority of the distances were between 1 and 3 inches.
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Experiment kinmodel_03: Equipartition of energy and the direction of disorder to (increasing or decreasing)
NOTE: The program is not currently set up to run the experiment as given here. See the alternative, a few lines below.
Observe the first several seconds of the simulation at the 'slow' default speed. You will see how the particles initially are all moving in or very close to the x direction, with little or no y component. Note the x and y kinetic energies, displayed near the top of the screen.
• Observe what happens to the directions of motion of the particles as they start colliding.
• Observe what happens to the x and y kinetic energies.
• Is the system more organized at the beginning of the simulation or after a couple of minutes?
I cannot change the default speed.
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• If the x and y kinetic energies were averaged for 100 years, starting a few minutes after the simulation began, which do you think would be greater?
I cannot change the default speed
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Run the simulation in this manner several times, and each time determine how long it takes before the total y kinetic energy is first greater than the total x kinetic energy. Report your results.
Now take some data:
• Running at the fastest default speed, stop the simulation with the pause/break key every few seconds, keeping your eyes closed for at least 2 seconds before stopping the motion n order not to bias your results.
• Write down the x and y kinetic energies each time, rounding to the nearest whole number.
• Do this at least 30 times.
• Find the average of all your x and all your y kinetic energies.
Give your data and your results:
I cannot change the default speed
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• Do you believe the difference in the averages is significant, in that the direction that has the higher average will always tend to have the higher average every time the simulation is run?
I cannot change the default speed
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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.
xyyyxxxxyyyyxxxyyyxyxxyyxxyyxxyxxxyyyyyyxxyxyxxyyyxyxx
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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).
1900, 1430 x
1700, 1500 x
1960, 1288 x
1900, 1348 x
1660, 1590 x
1598, 1651 y
1300, 1948 y
1445, 1600 y
1851, 1397 x
1751, 1498 x
1679, 1569 x
1400, 1846 y
1438, 1811 y
1435, 1814 y
1616, 1633 y
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What is the greatest KEx value you observed and what is the least?
1960, 1400
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What is the greatest KEy value you observed and what is the least?
1948, 1397
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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?
1680, 112
1672, 110
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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.
No, there is very little difference between the x and y values there means or there standard deviation.
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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.
31 times
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• Enlarge the square to a 1-inch by 2-inch rectangle and repeat.
18
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• Enlarge to a 2-inch by 2-inch square and repeat.
11
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• Enlarge this square to a 2-inch by 4-inch rectangle and repeat.
3
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• Enlarge to a 4-inch by 4-inch square and repeat.
1
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• Mask all but 1/4 of the screen and repeat.
0
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• How long do you think it would take, on the average, for 1/4 of the screen to become completely empty of particles?
I don’t think that ¼ of the screen would ever be empty of particles.
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• 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 1/2 of the screen would ever be empty of particles.
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• 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?
It would be a very long time since I don’t feel as if all the particles would ever end up on one side of the closet.
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Experiment kinmodel_05: The probability that a particle's speed will occur in a given range
NOTE: This experiment is pretty much redundant with a previous one and is to be OMITTED.
The default settings will not work with this experiment. This time when you run the program you need to choose to customize the settings. For everything but the numbers of different particles and their masses, and the 'yellow' marker, you may use the defaults.
For the number and masses of particles:
• When asked for the number of particles of type 1, enter 28. When asked for the mass of this type give 1.
• When asked for the number of particles of type 2, enter 1. When asked for the mass of this type give 10.
• When asked for the number of particles of type 3, enter 1. When asked for the mass of this type give 1.
Regarding the 'yellow' marker:
• You will choose the minimum and maximum speeds which will result in the particle being 'painted' yellow. This will allow you to observe the proportions of the particles in different velocity ranges.
If you wish you may also adjust the speed factor, which has default value 3. If you want the simulation to slow down to 1/3 the pace, you can enter 1 for the speed factor. If you want the simulation to go as fast as practical for the other default setting, you could use a speed factor up to 5. Only the pace of the simulation is affected by the speed factor; the speeds displayed on the screen are not affected.
Now try to observe the numbers of particles in various ranges:
• Run the simulation and use a 'yellow' range of 3 to 6 and attempt to observe the proportion of the particles falling within this range. You will be able to get a fairly good idea of the proportion, but it will be hard to get a really good estimate unless you repeatedly pause the program and count the 'yellow' particles.
• Run the simulation using a 'yellow' range of 4 to 4, which will give mark only particles whose velocity is 4. Determine to reasonable accuracy the average percent of particles with this velocity.
• Repeat for velocities 0, 1, 2, 3, 5, 6, 7, 8, 9 and 10.
What are the percentages corresponding to each of these velocities?
What therefore do you think is the average particle velocity?
Experiment kinmodel_06: The connection between relative particle mass and average speed; equality of average kinetic energies
Using default settings, answer the following:
• What do you think is the average speed of the dark blue particles as a percent of the average speed of the green particles? (you might, for example, observe how long, on the average, it takes a particle of each color to move a distance equal to that across the screen)
The dark blue particles are about ¼ the average speed as the green particles,
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• What do you think is the average speed of the red particle as a percent of the average speed of the green particles?
The red particles is about 1/10 the speed of the green particles.
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• A blue particle is 4 times more massive than a green particle. How do you think its average KE therefore compares with the average KE of the green particles?
I guess they would be similar since the speed of the green particles is greater.
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• A red particle is 64 times more massive than a green particle. How do you think its average KE therefore compares with the average KE of the green particles?
Similar since the speed difference would make up for the weight some.
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&#Very good data and responses. Let me know if you have questions. &#