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course Phy 232
7/8 630pm
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?
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It seems to take about 2-3 second for the particle to move across the screen at the speed of 10. If the speed is 5, it should therefore take twice the amount of time, so somewhere between 4-6 seconds. Based on my observations, this is correct.
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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?
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It varies, but I thought that it mostly took about 3-5 seconds before the particle collided with another. I think about 10 % of the collisions are less than a second. About 30% are less than 2 seconds, 50 % are less than 4 seconds, and 10 % are 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?
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I think about 10 % of the time the red collides with another in less than a second, and 25 % are less than 2 seconds. About 35 % are less than 4 seconds, and about 30 % are less than 10 seconds. It would take the red about 20 seconds on average to reach one end of the screen to another. Based on my observations, the average speed is about 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.
The blue particle differs in mass than the rest. They all seem to differ in mass since they have different reaction when they collide together.
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.
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I believe they spend most of the time in medium speed, and then slow speed and then fast speed. So they spend about 50 % of their time at medium speed, 30 % at slow speed, and 20 % at fast speed.
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Draw a histogram (a bar graph) of your estimates. Describe your histogram.
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The largest bar is the one for medium speed, which is in the middle. Then the slow speed on the left is the next highest, and the fast speed.
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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.
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For the first velocity range, I believe about 5 % is spent, and then 10 % in the next velocity range, and 10 % in the third. About 12% in the fourth, 15 % in fifth, 15 % in sixth, and 13 % in seventh. About 8 % in eigth, 7 % in ninth, and 5 % in tenth.
My histogram has the highest bars in the middle, and are smallest at the ends. They start small, then increase towards the middle, then decrease towards the end.
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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?
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The speed that is most likely is 7, which is right in the middle. About 25 % of the area underneath the curve corresponds with speeds within
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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?
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It spent about 15-20% of the time at less than half of its most likely speed.
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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.
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5,6,7,7,5,2,4,1,1,7,7,7,4,2,7,6,10,9,5,5,6,8, 3, 13, 4, 2, 2, 5, 2, 4, 4, 4, 6, 4, 7, 5, 6, 3, 8, 5, 3, 9, 2, 1, 1, 6, 4, 10, 8, 10, 9, 4, 4, 7, 4, 2, 5, 5, 2, 10, 5, 5, 4, 6, 6, 4, 5, 8, 6, 4, 6, 7, 7, 7, 4, 5, 6, 8, 5, 5, 7, 5, 1, 7, 3, 9, 6, 3, 8, 8, 4, 7, 5, 3, 1, 3, 3, 7, 7, 6
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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.
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1: 6
2: 8
3: 8
4: 16
5: 17
6: 13
7: 16
8: 7
9: 4
10: 4
11: 0
12: 0
13: 1
My histogram is pretty similar to the one I predicted earlier. The majority of the time is spent in the medium speed section, with the highest being around 5. The next majority is the slow section, and there are barely in the fast section.
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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.
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The distance between the walls that confine the particles is about 4 inches. The particle travels less than 1 inch before a collision about 20 % of the time, between 1 and 2 inches 40 % of the time, between 2 and 3 inches 25 % of the time and between 3 and 4 inches 15 % of the time. The histogram starts small, then gets high in the middle, then gets smaller again.
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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).
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I obtained my results by using the piece of paper with 15 folds and following the line on the green ball as it collided. I then figured out in inches the distance traveled.
1: 5
2: 5
3: 5
4: 4
5: 4
6: 0
7: 0
8: 3
9: 1
10: 0
11: 1
12: 2
13: 0
14: 0
15: 0
Since these are not inches, I rounded these results to inches:
0: 4
1: 14
2: 4
3: 3
4: 1
5: 3
6: 0
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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.
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The histogram of my results are very similar except for a peak in the 1-2 inch area. The histogram I would expect form a large number of observations would be more equal between the intervals, but the lower the distance will probably always be a little bit higher. The only thing wrong with my prediction is I thought the 0-1 inch area would have less than it did.
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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?
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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?
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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:
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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?
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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.
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xxyxxyyxyyyxxxxxyyxyyxxxyyyxxyxyyyxyyxyyxyxyxyyxyy
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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).
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yyyyyxyyyyyyyxxyxyxyyyxxxxxxxxxxxyyyyyyxxxyyxxxxyy
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What is the greatest KEx value you observed and what is the least?
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The greatest value I observed was 2612 and the least was 1332.
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What is the greatest KEy value you observed and what is the least?
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The greatest value I observed was 2201 and the least was 921.
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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?
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KEx Mean: 1972
Standard Deviation: 256
KEy Mean: 1561
Standard Deviation: 256
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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.
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I do not think they differ significantly, sometimes they are very close, and over the 50 intervals the values changed back and forth so one would be greater than the other. They also had the same 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.
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The square is empty about 15-20 % of the time.
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Enlarge the square to a 1-inch by 2-inch rectangle and repeat.
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The square is empty about 10 % of the time.
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Enlarge to a 2-inch by 2-inch square and repeat.
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The square is empty about 5 % of the time.
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Enlarge this square to a 2-inch by 4-inch rectangle and repeat.
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The square was empty maybe 1% of the time.
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Enlarge to a 4-inch by 4-inch square and repeat.
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The square was never empty in the 2 minutes I was watching.
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Mask all but 1/4 of the screen and repeat.
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The screen was empty less than 5% of the time.
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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?
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It would take the screen about 2 minutes, maybe more, on average to become completely 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?
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It might take, on average, about 5 minutes for the screen to be completely 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?
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I think it could take years, since it takes 2 minutes for a 1/4 of the screen to be clear yet the closet is 100 millioin times larger than the full screen.
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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)
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The average speed of the dark blue particles seem to be about 60% of the average speed of the green particles, based on my observations.
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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?
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I think the average speed of the red particle is about 15 % of the average 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?
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Its average KE would be only a little more than the green particle, because you have to take into account the v^2 in the KE equation and that velocity of the green is higher so that slightly balances the fact that the blue is heavier.
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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?
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The red particles average KE would be close to the green particle, because the green particle is about 8 or 9 times as fast, so while the velocity will be squared and therefore be higher for the green, the mass would be 64 times greater and they would balance out.
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Experiment kinmodel_07: The development of empirical frequency vs. speed and frequency vs. energy histograms (more about order vs. disorder, with statistical order emerging from the disordered system)
NOT CURRENTLY ASSIGNED:
The frequency vs. speed, frequency vs. square root of energy, and frequency vs. energy histograms (it is left up to the student to determine which is which) are normalized to have a consistent total area. These distributions develop over time, eventually reaching a smooth distribution analogous to the Maxwell-Boltzmann distribution. This development occurs much more quickly if the settings are customized to encourage a maximal number of collisions.
Experiment kinmodel_08: Images of 2-dimensional collisions
NOT CURRENTLY ASSIGNED:
It is easy to customize the settings to obtain two large relatively slow particles. Any student who has watched air hockey pucks or billiard balls colliding will recognize the validity of the simulation.
If the particles leave 'tracks' then a 'snapshot' in which a single collision between the particles occurs will provide data sufficient to validate conservation of momentum.
Customized settings
NOT CURRENTLY ASSIGNED:
The following parameters can be controlled by electing to customize settings:
The number of particles (default setting is 30, maximum is 1000).
The number of different particle types (more than 9 different particle types is not recommended because of restricted number of colors).
The speed factor that determines how fast the particles move across the screen. If the speed factor is too great, particles may occasionally (or frequently, depending on how great) miss collisions. This is not a big problem unless data is being taken that assumes no 'misses'.
The radius of a particle (default radius is 1% the width of the square viewing area).
The proximity of the centers of the particles within which collision will occur (default is 5 particle radii). A greater value here will result in more collisions, other parameters being equal.
The minimum and maximum speeds defining a speed 'window'. Any particle whose speed is in this 'window' will be colored bright yellow. This range of speeds will be indicated by a yellow rectangle on one of the graphs.
Whether all the particles leave 'tracks' or not. The last two particles usually leave 'tracks'.
The number of iterations before the screen is cleared and the various graphs are updated. An iteration consists of the calculation and display of the position of every particle. A fairly small number allows the viewer to observe the evolution of the graphs, while a somewhat greater number permits observation of a significant number and variety of particle 'tracks'. If the number is too great the particle 'tracks' will be obscured.
The last two particles specified will have velocities indicated onscreen; the last of these particles will be sampled to obtain the velocity distribution shown at the right of the screen.
'Research' questions
For which particle speeds is the time between collisions likely to be greatest, and for which will it be least?
Initial particle speeds are uniformly distributed. After a short time a specific nonuniform distribution of speeds takes over. How long does it take before the contribution of the initial uniform distribution to the graphs and histograms displayed on the screen become indistinguishable? How will the shape of the graph differ from the ideal distribution during the transition?
For the default settings, what is the 'peak' particle energy? What is the 'peak' velocity of the sampled particle?
A narrower speed range near the peak of the speed distribution can result in more instances of 'yellow' particles than a wider speed range away from the peak. At each possible integer speed v, it is possible to define a speed range (v0, vf) with v at the midpoint of that range, such that the average number of 'yellow' particles will be the same as for the 'unit' range around the peak of the distribution. The 'unit' range is a velocity range of width 1 unit centered at the 'peak' velocity.
What does it take to get a massive molecule surrounded by low-mass particles moving fast?
Does the presence of an even more massive particle give a medium-mass particle, surrounded by a greater number of low-mass particles, an advantage in achieving greater speeds? Does the presence of a more massive particle affect the energy distribution of the medium-mass particle?
At an advanced level: Derive Maxwell-Boltzmann distribution in 2 dimensions and compare the the empirical distribution.
More information on this model."
&#Your work on this lab exercise looks good. Let me know if you have any questions. &#