course Phy 202
3/5 around 11:30
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?
• 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?
For which particle speeds is the time between collisions likely to be greatest, and for which will it be least?
- It appears as though the faster the particle, the least amount of time between collisions. The sole red particle is by far the slowest, and is barely in collisions.
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 doesn’t take long for these particles to lose their uniform distribution of speed. After a few seconds it becomes obvious that the particles are moving at different speeds. The graph will no longer be as predictable I’d say. With all the particles moving at different speeds…
For the default settings, what is the 'peak' particle energy? What is the 'peak' velocity of the sampled particle?
- After looking at the Billiards simulation, it appears as though the peak KE is around 2000 Joules. After watching the KE scales, I’ve seen it jump into the high 1900s, but nothing higher than that. And after watching for a while, I saw the speed get no higher than 11m/s
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?
- The massive red particle is, for the most part, constantly surrounded by the low-mass/fast-moving green particles. The massive particle appears to increase the speed of the low-mass particles when the two collide.
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?
- If an even more massive particle was thrown into the mix, I think it would definitely increase the speeds at which the low-mass particles would be moving. As for the medium-mass particles, I feel as though they would act in a similar way as the original low-mass particles did with the original massive-particles. So I would say that the even more massive particle would affect the energy distribution.
At an advanced level: Derive Maxwell-Boltzmann distribution in 2 dimensions and compare to the empirical distribution.
- The Maxwell distribution, according to kinetic theory, is a way to derive the speeds of molecule in a gas. After looking at the Billiards simulation, the highest speed I observed was 11m/s, so that speed would most likely be down towards the end of the curve. The speed stayed most of the time within the 3-5 range, so I would put 4 as the peak of the Maxwell curve. I am not 100% sure how to derive the actual numerical distribution, but after reading over the information in the book, would I need to know the temperature of the system, what the actual particles in the simulation are, and things of that nature? And from what I read in the book, I don’t think I see where there is an equation that solves for the distribution.
Derivation of the distribution is way beyond the scope of this course; you would see that in an advanced undergraduate course in physics. You appear to have a good understanding, appropriate to the level of this course.
Good work.