#$&*
course Phy 232
6/15/12 submitted around 12:51PM
Experiments and ActivitiesPreliminary Observation
Run the program kinmodel, accepting all defaults by using the 'Enter' key to answer the prompts on your computer (the computer will then automatically pick the selection with the asterisk) and observe the particles or 'billiard balls' bouncing around the screen and off one another.
Watch the KEx and KEy values as they change with each collision, representing the total x and y kinetic energies of the particles.
Watch the 'red' particle for a couple of minutes, estimating the average time between its collisions and its average speed (one of the speeds given near the top of the screen corresponds to that of the 'red' particle--which is it?).
o The red particles average velocity is very low compared to the green particles. The time between collisions of the red particle is very high compared t the green particles.
Watch the 'blue' particle, and speculate on what property of this particle is different from that of the other particles.
o The blue particle has a greater mass and it would take more of the green balls hits to make it move.
Watch as the 'red' particle sometimes turns yellow. What causes this? What property does the particle have when it is yellow?
o In my program the red particle does not turn yellow.
What might the graphs represented at the right of the screen represent?
o The graphs do not work in windows 7 64-bit operating system
Strike the 'S' key to stop the simulation, and if you are done give the appropriate response to the prompt to quit the program. CTRL-ALT-DELETE will also stop the program, but if you're not careful it will reboot your computer so avoid that option if you can.
Before reading further email your instructor with your best answers to these questions. There are two good reasons for not reading ahead: If you get your answers by reading ahead your instructor will be able to tell, and if you read ahead you won't learn as much.
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 kinmodel at the default settings--simply hit the 'Enter' key for each option presented.
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.
o 43% slow, 45.6% medium, 10.8% fast
Draw a histogram (a bar graph) of your estimates. Describe your histogram in your writeup.
o The histogram spend about the same time in the slow and fast range and then about one fourth in the fast.
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.
o I estimate that the graph should have a bell curve shape to it.
Watch the red particle for long enough to estimate the percent of time it spends colored yellow.
Watch the whole simulation to see what average percent of the particles are yellow at a given time (there are 30 particles, including the blue one).
How would you expect the answers to these two questions to compare?
o In my program the red particle does not turn yellow.
Watch the number corresponding to the speed of the 'red' particle.
Close your eyes for a few seconds at a time and open them suddenly, and each time write down the velocity of the 'red' particle immediately when 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.
o The program does not show me the velocity of the red particle.
Observe the 'blue' particle for at least 5 minutes, monitoring its speed to see if it ever reaches 3 or more.
Based on your observations and your experience with other distributions, sketch a histogram of the speed distribution for the 'blue' particle, 'rounding' your results to the nearest .25 (i.e., use 0, .25, .5, .75, etc. for your velocities).
o The windows program does not show speed of the blue particles.
Stop the simulation and quit it. Then run it again and watch the 'blue' graph at the right. Sketch this graph every minute or so, for about five minutes. Describe how the graph develops, and describe what you think it will look like after a long time.
What do you think this 'blue' graph represents?
o The windows program does not show any graphs
What do you think the 'yellow' rectangle is for, and what does the graph tell you in relation to this rectangle?
o The windows program does not show any graphs.
Experiment kinmodel_02: Mean free path; mean time between collisions
It is possible to observe a chosen particle (the red or the blue particle in the default simulation) for its mean free path between collisions. This observation becomes more involving if, for example, the observer is 'rooting' for the red particle (in the default simulation) to collide with the blue particle. The tracks left by these particles also provide a record of the path between collisions.
First observe the 'red' 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 1/2 inch between collisions, the percent of the time the distance rounds off to 1 inch, the percent of the time the distance rounds to 2 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.
o 80% travels less than ½ inch between collisions, 15% travels between ½ and 1 inch, 5% travels more than 1 inch.
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/Break key on your computer, stop and start the particle motion as required in order to observe the distances traveled by the 'red' particle (the computer will stop when the key is depressed, and can be restarted using the 'Enter' key). 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).
o I measured the distance between each collision. 80% travels less than ½ inch between collisions, 17% travels between ½ and 1 inch, 3% travels more than 1 inch.
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.
o The histogram is going to look like a decaying exponential curve. They compare very close to the previous predictions.
Experiment kinmodel_03: Equipartition of energy and the direction of disorder to (increasing or decreasing)
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?
o At the beginning the system particles are moving in random directions and it does the same thing a couple minutes after.
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?
o I think that they would both be the same.
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.
o 1530.9 is the x average and 1401.1 is the y average
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?
o I dont believe the difference in the averages is significant.
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.
o There is 5% of the time this square is empty
Enlarge the square to a 1-inch by 2-inch rectangle and repeat.
o The square had some particle at all times
Enlarge to a 2-inch by 2-inch square and repeat.
o The square had some particle at all times
Enlarge this square to a 2-inch by 4-inch rectangle and repeat.
o The square had some particle at all times
Enlarge to a 4-inch by 4-inch square and repeat.
o The square had some particle at all times
Mask all but 1/4 of the screen and repeat.
o The square had some particle at all times
How long do you think it would take, on the average, for 1/4 of the screen to become completely empty of particles?
o 55 days
How long do you think it would take, on the average, for 1/2 of the screen to become completely empty of particles?
o 24353 years
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For a 4 inch by 4 inch square to be empty all 16 one-inch squares must be empty. If the probability of a one-inch square being empty is .05 (per your observation of the 1-inch square) then the probability of all 16 squares being empty is .05 ^ 16, which would be on the order to 10^-21. So it's not surprising that you didn't observe this.
Assuming that 'empty' means no particles in the region for, say, a millisecond, then is would take on the order of 10^21 between this observation. That's 10^18 seconds, or about 10^8 * 1 billion years.
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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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There's always a chance, but in this case the chance would be vanishingly small.
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o There is no chance for this to happen in the lifespan of the universe.
Experiment kinmodel_05: The probability that a particle's speed will occur in a given range
I cannot do this section because the graphing features or velocity features do not work on windows 7 64bit operating system.
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 percent of the time is the blue particle at each of the velocities 0, 1, 2, and 3?
o I cant measure the velocity of the blue particle
What therefore do you think is the average velocity of the blue particle?
o I cant measure the velocity of the blue particle
The blue particle is 10 times more massive than the other particles. How do you think its average KE therefore compares with the average KE of the other particles?
o I think its average KE is greater than the other particles.
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)
The frequency vs. speed, frequency vs. squart 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?
The slowest heaviest particles have the greatest and the lightest fastest particles have the 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?
The program does not allow me to see the graphs.
For the default settings, what is the 'peak' particle energy? What is the 'peak' velocity of the sampled particle?
I cant tell since the program does not record individual particle velocities.
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?
It would take many collisions in the same direction.
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?
Yes by getting hit by a more massive particle the particle will move faster and it would 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.
We dont know what the empirical distribution since the program didnt display graphs.
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&#Good work. Let me know if you have questions. &#