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course phy122
2/8 3Please let me know if I need to do any revisions.
Introduction
The 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?
It takes less than one second for the green dot with a tracer to cross the screen. It should take twice as long for the object to cross the distance if its speed is 5. Yes, that is about what I observed.
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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?
The green particle hits another particle an average of 58-60 times in a one minute period. I would estimate that approximately 20% of collisions are less than one second. It is really hard to determine this based on the fact that the green dot moves so quickly around the screen.
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Good.
Do note that your responses need to go into the space below the question, which below the **** and above the #$&*.
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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?
Less than one second about 4%, less than 4 seconds about 16%, less than 10 seconds: 40 %; it takes the red particle about 26 seconds to move a distance equal to the height or width of the screen. The average speed of the red particle 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 royal blue particle is slightly slower than green. I would assume that it is heavier than the green and lighter than the red particle.
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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Percentage of time spent at slow: 20 %
Percentage of time spent at medium: 60%
Percentage of time spent at fast: 20 %
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Draw a histogram (a bar graph) of your estimates. Describe your histogram.
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My histogram will have a small column on the left hand side of the screen containing 20%, a tall column in the middle containing 60%, and a small column on the right hand side of the screen containing about 20 %.
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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 histogram will start on the left hand side of the screen, with shorter columns gradually increasing in size until they reach a tall tower in the center, and then the columns will start to gradually decrease in size.
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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 most likely seen is 1383 (this is somewhere in the midrange). The percentage underneath the curve is about 80 percent of the speeds within one unit of my most likely speed. The speed associated for ½ as high as the maximum is: 826. The speed that is ½ as high as half as high as maximum is: 413
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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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The percentage of time that the green particle spends at speeds more than 2 units is 13.3%; it is moving at less than half its most likely speed 7 % of the time.
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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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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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This histogram is similar to my previous assumed histogram, however now that I actually used the data obtained my histogram now is going to show a heavier concentration in the 4-9th column (bars).
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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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%of time that the particle travels less than 1 inch between collisions=21%
% of time distance is between 1 and 2 inches: 28%
% of time between 2 and 3 inches:37%
The distance between the walls horizontally is:17 cm
The distance between the walls vertically is:15 cm
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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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The data was obtained by pausing and starting the kinetic model 30 different times in order to measure the distance between collisions with the green particle with a tracer. The distance was then measured from one point to the other. The data was then recorded into a histogram.
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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 percentage in column one shows how many distances were from the 0-7/8 inch range. Column 2 dictates how many measurements were in the 1-1 7/8 inch range. Column 3 dictates how may measurements were in the 2 -2 7/8 inch range. Column four dictates how many measurements fell into the 3-3 7/8 inch range. Column 5 dictates how many measurements fell into the 4-4 7/8 inch range. Column 6 dictates how many measurements fell into the 5-5 7/8 range. Column 6 dictates how many measurements fell into the 6- 6 7/8 inch range.
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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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yyyxyyyxyxyxyxxyxxyxyxyxyxyyyxyyxyyyyyyyxyyyyxyyyy
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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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xyyxxxxxxxxxxxxxxxxyxyyyxyxxyxxyyxxyyyxyyyxxxxxx
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What is the greatest KEx value you observed and what is the least?
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Greatest x: 1699
Least x:1059
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What is the greatest KEy value you observed and what is the least?
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Greatest y: 1635
Least y: 999
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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: 1379(mean), 64 standard deviation
keY 1317 (mean), 63.6 standard deviation
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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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It does differ but not significantly. The difference is by 62. The standard deviation only differs by 0.4.
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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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32
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Enlarge the square to a 1-inch by 2-inch rectangle and repeat.
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9
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Enlarge to a 2-inch by 2-inch square and repeat.
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5
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Enlarge this square to a 2-inch by 4-inch rectangle and repeat.
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0
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Enlarge to a 4-inch by 4-inch square and repeat.
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0
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Mask all but 1/4 of the screen and repeat.
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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?
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I am unable to make this prediction, however I have been watching this simulation all morning and Ό of the screen has not been clear as of yet.
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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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I dont think this would happen.
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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 have been watching this simulation all morning, and all of the particles have not gone to one side at any point this morning. I dont think that it would happen.
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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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Blue: 8 sec
Green: 5 sec
Im not sure that I understand this question.
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If for example the blue particle is moving half as fast as the green this would be 50%.
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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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Red: 48 sec
Green: 5 sec
Again, I am not sure that I understand this question, but I would be glad to answer it professor if you wouldnt mind rephrasing it.
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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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I think because it has greater mass, that it should have more kinetic energy than the green particles.
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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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I also think that the kinetic energy of the red particle should be greater than the green particle. I think of the particle as a wrecking ball. I am going to equate the green particle to a lesser weighted wrecking ball and the red to a greater weighted wrecking ball. The kinetic energy should be greater with the red particle in this example. Im hoping that I have not confused myself with potential energy. I do however have hesitations in my answer. The longer I think of this the more I think about how slowly the red particle was moving than the green particle. In that perception, I would assume that the red particle would have less kinetic energy.
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Kinetic energy is 1/2 m v^2.
A particle with greater mass and less velocity could have the same KE as a particle with a lesser mass and a greater velocity.
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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?
I think that the time between collisions is likely to be the greatest for the green particle. I think that it will be the least for the red particle, because of its mass and slowness of speed, collisions are likely to occur at an increased rate.
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?
It only takes a matter of seconds before the histograms change. Im not sure that I completely understand this question.
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.
Peak velocity can be calculated by taking the distance covered and dividing by the time it takes to cross that distance. For the red particle, it crossed the 15cm vertical distance in 26 seconds. So 17/26=0.654cm/sec
What does it take to get a massive molecule surrounded by low-mass particles moving fast?
Multiple collisions.
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?
Absolutely to both!
At an advanced level: Derive Maxwell-Boltzmann distribution in 2 dimensions and compare the the empirical distribution.
More information on this model.
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Well done. Check my notes.
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