#$&*
course Mth 163
3-20-13 3:09
http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
019. `query 19
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Question: `qexplain the steps in fitting an exponential function to data
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
With at least two points of data, you can solve two simultaneous equations in y=Ab^x.
confidence rating #$&*:
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Given Solution:
`a** If you have two points you can solve the simultaneous equations:
• Substitute the coordinates into the form y = A b^x and solve the two resulting equations for A and b.
• You could alternatively use the form y = A * 2^(k x) or y = A * e^(k x), in which case you would solve for A and k.
If you have a more extensive data set you can use transformations.
For exponential data you plot log(y) vs. x. If the graph is well approximated by a straight line then you get an exponential function.
Then since the graph is a straight line, you can find its equation using using either slope and vertical intercept, or two points on the line.
If the slope of a y vs. x graph is m and the vertical intercept is b then the function is y = m x + b.
However in this case the graph is not of y vs. x, but of log(y) vs. x.
So if the slope of your graph is m and the y intercept is b, the function is log(y) = m x + b.
This equation needs to be solved for y:
You invert the transformation using the inverse function 10^x, obtaining 10^log(y) = 10^(mx+b).
10^log(y) = y, by the definition of the logarithm, and
10^(mx + b) = 10^(mx) * 10^b, by the laws of exponents.
Thus
• y = 10^(mx) * 10^b,
where m and b are just the numbers (slope and vertical intercept) that you determined from your graph.
Note that if a power function fits the data then log y vs. log x will give a straight line so that log y = m log x + b. In this
case our solution will be y = 10^b * x^m, a power function rather than an exponential function. **
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Self-critique (if necessary):
ok
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Self-critique rating: ok
If you understand the assignment and were able to solve the previously given problems from your worksheets, you should be able to complete most of the following problems quickly and easily. If you experience difficulty with some of these problems, you will be given notes and we will work to resolve difficulties.
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Question: The graph of log(y) vs. log(x) has slope 2.5 and vertical-axis intercept 4.
What is the equation relating log(y) to log(x)?
What is the equation relating y to x?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
The equation is log(y)=2.5(log(x))+4
Assuming that the inferred base is 10, then this: log(baseb)x
Log(base10)(y)=2.5log(base10)^(x+4) x=log(baseb)(b^x)
Y= 10000*2.5
@&
Most of your steps are right, but you don't quite get the right expression at the end. I expect this is a typo, but just to make sure:
If log(y) = 2.5 log(x) + 4 then
10^(log(y)) = 10^(2.5 log(x) + 4)
so
y = 10^(2.5 log(x)) * 10^4
= 10 000 * (10^(log(x)) )^2.5
= 10 000 * x^2.5
*@
x=b^log(baseb)x
Assuming log(x) is the natural log, then this:
Y=54.5982x^2.5
@&
This is right, which is what makes me thing your last answer was a typo.
*@
confidence rating #$&*: 2
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Question: The graph of log(y) vs. x has slope 2.5 and vertical-axis intercept 4.
What is the equation relating log(y) to x?
What is the equation relating y to x?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
Log(y)=2.5x+4
10^(log(y))=10^(2.5x+4)
Y=10^2.5x*10^4
Y=316.2277x*10,000
@&
y=10^2.5x*10^4 needs a little grouping:
y=10^(2.5x)*10^4
so that
y = 10 000 * (10^2.5)^x = 10 000 * 316^x,
not 10 000 * 316 x.
*@
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Self-critique rating: ok
@&
You appear to have a couple of errors among a lot of right steps.
If you're very careful about applying the laws you know, you'll avoid these errors.
Check my notes and let me know if you have questions.
*@
confidence rating #$&*: 2
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Self-critique (if necessary):
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Self-critique rating:
course PHY 122
3/20/13 4PM
Preliminary ObservationRun 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?
At a speed of 10, the time required is about 1 second. At 5, about 2 - 2.5 seconds.
How frequently does that green particle collide with other particles?
The frequency of collisions varies greatly. There are instances of several collisions occuring per second, but at times, the ball will move for several seconds with no collisions.
What percent of the time intervals between collisions do you think are less than a second? What percent are less than 2 seconds? What percent are less than 4 seconds? What percent are less than 10 seconds?
I would guess around 30% occur in less than 1 second, 50% are less than 2. Another 15% occur within 4 seconds. I never observed a 10 second interval without a collision, but I assume that under the perfect circumstances it could occur.
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?
15% less than 1 second, 30% less than 2, 30% less than 5. I did observe a 10 second interval with the red ball when it was ""glued"" to the side of the windows traveling up the screen. I would estimate that maybe 2-3% of collisons occur within 10.
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?
I would guess around 12 seconds.
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?
The red particle moves slower than the green particle at a speed of 1, so I would say around .5.
Watch the 'blue' particle, and speculate on what property of this particle is different from that of the other particles.
The blue particles clearly have a greater mass than the green particles, but significantly less mass than the red.
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.
The fastest speed I observed was 12. The ball spent spent most of its time in the first two ranges and rarely stayed over 2/3 of maximum for any length of time. I estimated the ball was in the 5-8 range more frequently, but for less time. My time histogram had three bars, the first corresponding to about 50%, the second 40%, and the last 10%.
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.
By my observations, the time spent in the 0-.1, .1-.2, .3-.4, and .4-.5, were fairly equal. For .3-.4 and .4-.5, the ball stayed at that speed for less time when it achieved it, but at a greater frequency. .2-.3 was the next interval the particle spent the most time at, but was a little less than the first group. The ball spent little time at each of the higher intervals, though .5-.6 was probably a little more than the highest speeds.
20
15 || || || ||
10 || || || || || ||
5 || || || || || ||
0 || || || || || || || || || ||
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0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0
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?
A speed of 5 and 6 were the most likely. About 40% of the time the speed would be within 1 unit of 6. A speed of around 7.5 would be half as high and about 9 would be half as high as that.
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?
Probably between 25 and 30 percent of the time. The ball frequently is slowed to a speed of 1, 2, or 3, and often stays there for several seconds at a time. The happens frequently when the ball is moving near the corners, or travels parallel to on of the sides of the window.
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.
My percentages were,
Speed Percent
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0 1
1 9
2 16
3 14
4 13
5 13
6 10
7 7
8 6
9 7
10 1
11 1
12 1
13 1
Overall, I feel this corresponds fairly close to my observations. While the frequencies of 5 and 6 are a little less than 2 and 3, I felt during my initial observations that the ball was staying at those speeds for a great amount of time. My initial belief that the frequency of the 5 and 6 speeds would cause the amount of time spent at those speeds to be equal to the slower speeds may have been inaccurte. The fastest speeds were certainly less represented, and after 9, are practically non-existant. My initial percentage estimates for these was probably a little high.
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.
0-1 27
1-2 21
2-3 11
3-4 10
4-5 8
5-6 4
6-7 6
7-8 9
8-9 4
I obtained these results by stopping and starting the program as instructed, and measuring the distance the ball traveled using the traced line as a guide.
These results were fairly close to what I assumed would happen. Move of the collisions occured within a couple of inches. I was surprised how frequently the ball traveled for several inches before colliding. I assumed there would not be more than one or two that would make it more than 3-4 inches. As in previous questions, when the ball was moving along an edge, or in a corner, the distance started to add up.
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.
xyyyyyxyxyxyyxxxxxyxxxyxyxxyyyxyxxxxxyxyxyyxxxyxyyxyxyxxxyyxxy
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).
yxyyyyxxxxyxxyyyxxxxyxxyyxxyxxyyxyyxyxxyyxyxyyyyxyyyxyyxyyyyyxx
What is the greatest KEx value you observed and what is the least?
1707, 843
What is the greatest KEy value you observed and what is the least?
1662, 881
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?
KE_x_mean = 1275
KE_x_deviation = 173
KE_y_mean = 1272
KE_y_deviation = 156
Do you think the mean KEx value differs significantly from the mean KEy value? There is a difference. By 'significantly', we mean a difference that
seems greater than what would naturally occur by chance statistical variations.
No. The difference between the two is already neglibible, and would probably decrease more with a greater number of data points.
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.
I observed 27 instances of the square being empty in 2 minutes. I would estimate that in none of the instances was the box empty for more than 1/2 of a second. I estimate that no more than 10% of the total time was the box empty.
Enlarge the square to a 1-inch by 2-inch rectangle and repeat.
For this trial, there were 7 times the box was empty. Not more than a couple percent at most.
Enlarge to a 2-inch by 2-inch square and repeat.
0%. The box was never completely empty
Enlarge this square to a 2-inch by 4-inch rectangle and repeat.
0%
Enlarge to a 4-inch by 4-inch square and repeat.
0%
Mask all but 1/4 of the screen and repeat.
0%
How long do you think it would take, on the average, for 1/4 of the screen to become completely empty of particles?
I don't think it would ever happen. At no point during any of these questions did I observe 1/4 of the screen empty.
Statistically, I suppose that it would have to happen at some point. Given that at no point during these experiments did I observe 1/4 of the screen empty, I would imagine it would take quite a long time.
Again, I don't think it would happen.
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?
Once again, it is probably statistically possible, but I would think it would a very long time (years? centuries?).
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)
About 50%.
What do you think is the average speed of the red particle as a percent of the average speed of the green particles?
10%
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?
If 50% of velocity is accurate, the KE would be approximately equal.
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?
Again, I think they would be approximately the same if my estimated velocity values are correct.
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.
??????????????????????????
Are there questions for this experiment, or is it just supposed to be performed.
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'Research' questions
For which particle speeds is the time between collisions likely to be greatest, and for which will it be least?
The fastest speeds would produce the least amount of time between collisions. The slowest would produce the greatest amount of time.
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?
I'm not sure if its a problem with my computer, or the application itself, but the histogram windows do not update. Just as a guess, I wouldn't think it would take very long before the initial state of the system is gone. Collisions occur too quickly and frequently for the velocities to remain uniformly distributed for any period of time.
For the default settings, what is the 'peak' particle energy? What is the 'peak' velocity of the sampled particle?
The highest KE values I observed were around 1700 for KE_x and KE_y. The highest velocity was 13.
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?
I assume this question is asking what does it take to get the massive particle moving fast. If this is correct, numerous collisions from high velocity low mass particles all striking the massive particle in a similar direction (all KE_x, KE_y) would increase the speed of the massive particle.
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?
Having a more massive particle would cause low mas particles to achieve higher speeds during collisions, meaning that it would be more likely that the medium mass particle would acheive a high speed.
At an advanced level: Derive Maxwell-Boltzmann distribution in 2 dimensions and compare the the empirical distribution."
&#Your work on this lab exercise looks very good. Let me know if you have any questions. &#