#$&*
course Phy 232
My answers are bulletpointed and noted by **** before and **** after.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?
• 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?
• 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?
• Watch the 'blue' particle, and speculate on what property of this particle is different from that of the other particles.
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• On average, it seemed that the green particle was traveling around a velocity of 5. It is hard to estimate the time since the green ball is constantly hitting the other balls. However, I believe that it takes around a second for the particle to travel across the screen at a velocity of ten. Based on this, it would take the particle roughly two seconds to go from one side of the screen to the other. After testing this idea, I believe that a particle at a speed of 5 it would take around two seconds to travel across the screen.
• After observing the simulation, I believe that the particle, on average, collides with another particle every one and a half seconds. Obviously it is extremely difficult to create these percentages, but rough estimates are as follows. I think that the particle collides with another particle within a second around 30% of the time and less than two seconds around 70% of the time. The particle should collide with another in less than four seconds about 95% of the time and within ten seconds 99% of the time. Even though ten seconds is a long time, I do not think we can say it is impossible for this to happen. In my short time studying the simulation, the particle failed to touch another particle for roughly 8.5 seconds. I know this is not greater than ten, but I believe it shows that these great lengths of time are plausible.
• It seems as if the red particle has a velocity of 1. On average, I believe it collides with another particle every two and a half seconds. I will estimate that the red particle collides with another particle in less than a second around 15% of the time. I believe it collides in less than two seconds around 50% of the time and within four seconds around 85% of the time. Lastly, I think that the red particle fails to touch another particle in 10 seconds around 98% of the time.
• I could not figure out what property the blue particle has that the others do not. I have noticed that this specific particle seems to do an orbit like thing, but it does not happen that often. Therefore, I do not think this is the specific trait of the particle, but it is worth noting.
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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.
• Draw a histogram (a bar graph) of your estimates. Describe your histogram in your writeup.
• 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.
• 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?
• 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?
• 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.
• 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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• I used a blue particle for my observation. The green particle seems to have a wide variety of speeds rather than the red particle which has a consistently low velocity. The graph of my histogram seems to fit a rather nice bell curve. There are more occurrences near the median speed and the farther you get from that average fewer occurrences at that velocity occur.
• The blue particle spends around 25% of the time in speeds less than a third its max, 50% in speeds between a third and two thirds its max, and 25% in speeds greater than two thirds its max velocity. These percentages, while being estimates, show the nice distribution of the velocities and the bell shaped curve I was discussing earlier.
• Even with 10 velocity ranges, the graph still has a similar shape, but is more distributed since there are more intervals. Therefore, .1 max speed would occur 2.5%, .2 max speed would occur 5%,.3 max speed would occur 10%,.4 max speed would occur 17.5%,.5 max speed would occur 30%,.6 max speed would occur 17.5%,.7 max speed would occur 10%,.8 max speed 5% and .9 max speed would occur 2.5%.
• Since the most likely speed .5 its max speed, I would estimate the area under the curve of .5-.1=.4 max speed and .5+.1=.6 the max speed. It seems as if from .4 max to .6 max the percent of the area is around 50%. Around a speed a little below .4 max and a little above .6 max, the curve is about half as high as the maximum.
• The green particle seems to spend about 15% to 20% of its time at speeds greater than 2 units but no more than 4 unit speeds over the average. The green particle moves at less than half of its average speed about 20% of the time.After doing this part of the experiment and recording 100 velocities of the green particle at various random times, there were no occurrences at speed 0, 4 occurrences at speed 1, 5 occurrences at speed 2, 9 occurrences at speed 3, 18 occurrences at speed 4, 29 occurrences at speed 5, 14 occurrences at speed 6, 9 occurrences at speed 7, 7 occurrences at speed 8, 3 occurrences at speed 9, and 2 occurrences at speed 10. After creating a histogram of my data, this is very similar to my earlier histogram. Even though my other one was based on percent to max speed this illustrates the same thing. This histogram has an obvious max at speed 5 which was similar to our other histogram which had a max at .5 or half max velocity. This histogram has a nice bell shaped curve to it as can be seen from the decreasing occurrences as the speed increases and decreases from the average speed of 5.
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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.
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. Use a ruler to measure distances traveled. Don't leave any distances out, because this would bias the sample. Observe at least 100 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).
• 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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• I think the average distance the green particle travels before colliding would be close to 2 inches. Therefore, I believe that between 0 to 1 inch would occur 15% of the time, from 1 to 2 inches 30% of the time, for 2 to 3 inches 30% of the time, 3 to 4 15% of the time and greater than 4 10% of the time. A histogram of this would show a bell curve with a max at the 1 to 2 and the 2 to 3. The graph would come together at a max in between these intervals.
• The dimensions of the box on my monitor is 5.5 inches by 5 inches .I used the pause and restart buttons to record my data which is given below to the nearest inch:
o 2,1,2,0,3,4,2,3,1,4,3,2,1,1,2,1,1,2,1,3,5,2,1,3,1,0,2,4,3,3,4,2,2,1,3,2,1,0,1,3,2,1,3,3,0,2,2,3,2,1,1,22,1,1,2,0,2,4,1,0,2,3,1,0,2,4,3,4,0,1,2,3,4,2,1,3,2,1,3,4,2,1,5,3,2,1,2,3,2,1,2,1,2,3,1,2,3,4,1
• My previous estimate of close to 2 inches seems accurate based on my data taken above. Just for my knowledge, the average of my data was 2.1 inches. Even though this was not a very large sample, it gives a decent estimate of the distance traveled in between collisions of the green particle. A histogram would once again have a bell curve. The most common length is 2 inches so that would have the highest bar while the 3 inch and 1 inch bars would be somewhat lower. The 4 inch bar and the 0 inch bar would be even lower than that since those distances showed up less frequently.
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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?
• 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?
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.
• Write down the x and y kinetic energies each time.
• Do this at least 30 times.
• Find the average of all your x and all your y kinetic energies.
• 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?
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.
At whatever pace you prefer, write or type about 50 observations of x or y. List them here.
xyxyyyxyxyyxyyxyyxyxxxyxyyxyyyyxyyxyxyxyyxyxyxyyxy
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. 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).
What is the greatest KEx value you observed and what is the least?
What is the greatest KEy value you observed and what is the least?
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?
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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• The 50 observations of x or y are as follows xyxyyyxyxyyxyyxyyxyxxxyxyyxyyyyxyyxyxyxyyxyxyxyyxy
• The 50 KEx and KEy xyxxxyxyyxyyxyxxxxxyyxyxyyxyyyxyyyxyyyyxyxyxyxyyxy.
• Greatest KEx value was 1726 and the least KEx value was 1033. Greatest Key value was 1594 and the least Key value was 981.
• Mean of KEx was 1521 while the standard deviation for KEx was 99.2.
• Mean of Key was 1487 while the standard deviation for KEy was 91.3.
• There is an obvious difference between the two, but I do not think it is out of statistical possibility. However, the fact that the screen or box size is not a perfect square might have something to do with the numbers not being closer together.
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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.
• Enlarge the square to a 1-inch by 2-inch rectangle and repeat.
• Enlarge to a 2-inch by 2-inch square and repeat.
• Enlarge this square to a 2-inch by 4-inch rectangle and repeat.
• Enlarge to a 4-inch by 4-inch square and repeat.
• Mask all but 1/4 of the screen and repeat.
• How long do you think it would take, on the average, for 1/4 of the screen to become completely empty of particles?
• How long do you think it would take, on the average, for 1/2 of the screen to become completely empty of particles?
• 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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• 1x1inch square was empty 32 times. The percent that the square was empty is roughly estimated to be as high as 30 to 40%.
• 1x2 inch square was empty 19 times. The percent that the square was empty is roughly estimated to be 20%.
• 2x2 inch square was empty 13 times. The percent that the square was empty is roughly estimated to be 10%.
• 2x4 inch square was empty 5 times. The percent that the square was empty is roughly estimated to be 5%.
• 4x4 inch square was empty 0 times. The chances of this square being empty seem impossible. However, you can never take this out of the realm of possibility and there is some, even if a fraction of a percent, chance of this occurring.
• One fourth of the screen had 0 times where it was empty. On my computer, one fourth of the screen is less than the 4x4inch square. Similarly to the above notion, the chances are small, but slightly greater than with the 4x4 square.
• It seems hard for me to come up with a time period when only studying this program for a short period of time. I guess I could say that it would take hours for this to happen.
• I believe it could take days or months for half the screen to become empty, if it will even occur in that amount of time.
• Even though the closet is much bigger than the simulation, it gives the same simple concept. Therefore, the time should be comparable to the amount based on the simulation for half the screen and for half 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)
• What do you think is the average speed of the red particle as a percent of the average speed of the green particles?
• 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?
• 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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• Based on a rough estimate, I believe the blue particle has an average speed 25% of the green particles average speed. This was calculated based on the weights and my ideas on the simulation.
• Based on a rough estimate, I believe the red particle has an average speed 1.6% of the green particles average speed. Once again, this was based on the masses and my ideas on the simulation.
• I assume all of the particles have the same KE and I have used this idea in the above questions. If they have the same KE, then 1/mass of a particle rather than green will give you the percentage speed of a particle in comparison to the green particles. This is why I said the blue particle moves at a speed .25 of the green because this will equalize the weight differences to have an equal KE.
• I assume all of the particles have the same KE and I have used this idea in the above questions. If they have the same KE, then 1/mass of a particle rather than green will give you the percentage speed of a particle in comparison to the green particles. This is why I said the red particle moves at a speed .0156 of the green because this will equalize the weight differences to have an equal KE.
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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
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
The following parameters can be controlled by electing to customize settings:
The number of particles (default setting is 30, maximum is 1000, which shouldn't be much of a restriction in the near future of PC's).
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?
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• The green particle has the shortest average time in between each collision. The red particle has the largest average time in between each collision.
• The particles lose their uniform distribution of speed in a very short amount of time. The graph will not have the ideal distribution shape and will become somewhat indistinguishable.
• The particle energy seems to peak at around 2000J and the max velocity seemed to be around 10m/s.
• The red particle is almost constantly surrounded by the low mass particles. In this case, I believe it takes quantity of low mass particles, which is what we have.
• A more massive particle would surely increase the velocity of the low mass particles. An even more massive particle would obviously affect the energy distribution of the system.
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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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&#Your work on this lab exercise looks good. Let me know if you have any questions. &#