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Random Motion. A note on usage:. The clicker slides in this booklet are meant to be used as stimuli to encourage class discussion. They are intended for use in a class that attempts to help students develop a coherent and sophisticated understanding of scientific thinking.
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A note on usage: The clicker slides in this booklet are meant to be used as stimuli to encourage class discussion. They are intended for use in a class that attempts to help students develop a coherent and sophisticated understanding of scientific thinking. They are NOT intended as items to test whether students are “right or wrong” or “know” the correct answerby one-step recall if enough cues are given. This has a number of instructional implications that are reviewed in general on the next four slides. The individual slides also contain annotations discussing their intended use.
Usage: 1 • Feedback One of the most important values of a clicker-response system is to provide instructors with some understanding of what students are thinking. Good clicker questions can be highly revealing (and surprising). But the critical fact is not that the students make mistakes but to use those mistakes to probe their thinking and find out why. This raises the importance of a rich subsequent discussion well above “letting the students know what the right answer is.”
Usage 2: • Student-student interactions The critical value for student learning occurs in what happens after a clicker question has obtained a mixed response from the students. The standard next cue is, “Find someone who disagreed with the answer you chose and see if you can convince them.” After a minute or two of discussion, a second click may show students having moved dramatically towards the correct answer. A brief call for who changed their answer and why can lead to a useful exchange. When they have not moved significantly, more discussion is called for.
Usage: 3 • Incompletely specified questions Some items have questions that are simple if idealized assumptions are made, subtler if they are not. Part of the discussion of these items are intended to include issues of modeling, idealizations, and hidden assumptions. • Questions where answers are not provided. In these items, the intent is to have students come up with potential answers and have the instructor collect them and write them on the board. Occasionally, especially at the beginning of a class, it may take some time before students are willing to contribute answers. It can help if you have some prepared answers ready, walk around the class, and put up the answers as if they came from the students. This can help students get more comfortable with contributing.
Usage: 4 • Cluster questions Some questions are meant to be used as part of a group of questions. In this case, resolving the answers to individual questions is better left until the entire group is completed. The value of the questions are often in the comparison of the different items and in having students think about what changes lead to what differences and why. • Problem solving items In these items (indicated by a pencil cluster logo), the intent is to have students work together to solve some small problem. After a few minutes, ask the groups to share their answers, vote on the different answers obtained, and have a discussion.
Air is a mixture of (mostly) oxygen (molecular mass 16) and nitrogen (molecular mass 14) gases. At room temperature, which molecules in this room have more kinetic energy (on average)? • Oxygen • Nitrogen • Same • Not enough information to tell
Air is a mixture of (mostly) oxygen (molecular mass 16) and nitrogen (molecular mass 14) gases. At room temperature, which molecules in this room have a greater speed(on average)? • Oxygen • Nitrogen • Same • Not enough information to tell
Imagine a sealed container containing a dilute gas. If the average speed of the molecules increased by 20%, what would happen to the pressure on the walls of the container? • It would remain the same as before. • It would increase by 20%. • It would increase by closer to 50% than to 20%. • It would increase to nearly twice as much as it was before. • You can’t tell from the information given.
In this simulation, a “walker” starts at 0 and steps left and right with equal probability. We will let it take N steps. If we release a lot of walkers from the origin at once, on the average, what will our distribution of particles look like? • There will be equal numbers near +N/2 and –N/2 • They will be mostly near 0 no matter how many steps you take. • It will peak at 0 and getting farther will decrease in probability. • There will be peaks at + and – values but not at +N/2 and –N/2; 0 will be less likely. Stp_RandomWalk1D.jar
In this simulation, a lot of “walkers” starts in 2D near 0 and step in a random directions with equal probability. As time grows, what will happen to the distribution of walkers? • They will form a “wave” – a ragged ring of particles moving outward. • They will be mostly stay near 0 no matter how long you wait. • It will peak at 0 and getting farther will decrease in probability, the distribution remaining mostly the same. • It will peak at 0 and getting farther will decrease in probability, the distribution getting wider with time.
The diffusion constant, D, for a small amount of molecules in a gas relates the flow of those molecules in one direction (J = number per unit area per second) to the gradient of the material’s density in that direction (number per unit volume, n) by Fick’s law . The diffusion constant in a dilute gas only depends on the mean-free path (average distance between collisions) of our molecules (λ) and their average speed (v0). A possible equation for D is
The motion of an E. coli bacterium is observed using a video camera. A log-log plot of the square deviation of a chosen bacterium as a function of time is shown below. It seems to have a different behavior for long times (t > 1 s) and short (t < 1 s). What might account for its behavior at short times (Region A)? • The bacterium is moving purposefully in response to some chemical gradient. • The bacterium is moving at random in response to the thermal motion of its environment. • The bacterium is constrained in some way. • The bacterium is using its flagella (which work like propellers)to move at a constant velocity • The bacterium is accelerating in response to a force in a fixed direction. • None of these behaviors are consistent with that part of the graph.
The motion of an E. coli bacterium is observed using a video camera. A log-log plot of the square deviation of a chosen bacterium as a function of time is shown below. It seems to have a different behavior for long times (t > 1 s) and short (t < 1 s). What might account for its behavior at long times (Region B)? • The bacterium is moving purposefully in response to some chemical gradient. • The bacterium is moving at random in response to the thermal motion of its environment. • The bacterium is constrained in some way. • The bacterium is using its flagella (which work like propellers)to move at a constant velocity • The bacterium is accelerating in response to a force in a fixed direction. • None of these behaviors are consistent with that part of the graph.
Flipping a coin If you flip a fair coin 4 times, which string are you more likely to get: (A) HHHH; (B) HTTH? • A • B • Equally probably • Not enough info to decide.
Flipping a coin If you flip a fair coin 4 times, which string are you more likely to get: (A) 4 heads; (B) 2 heads and 2 tails? • A • B • Equally probably • Not enough info to decide.
Flipping a coin You flip a coin 10 times. How many different sequences like HHTTHTHTTH (microstates) are possible? • 10! (= “10 factorial” = 10 x 9 x 8 x …x1) • 102 • 210 • 10 • 11 • Some other number
Flipping a coin You flip a coin 10 times. How many different results like 5 heads and 5 tails (macrostates) are possible? • 10! (= “10 factorial” = 10 x 9 x 8 x …x1) • 102 • 210 • 10 • 11 • Some other number
We use probability for flipping coins because • The world of the coin is fundamentally random. • The world of the coin is fully predictable given the starting conditions, but we can’t determine the starting conditions well enough. • We want to to be able to set good odds for our bets. • We want to be able to predict the future. • Some other reason.
How do we know that a coin is “fair” for flipping? • We buy a special coin from a trustworthy manufacturer. • We use any US coin because the US Mint explicitly makes all coins to be flipping fair. • We test it by making an infinite number of flips. • We test it by making a finite number of flips. • There is no way that we can know a coin is fair.
Suppose you thought a coin was fair and flipped it 30 times and it came up 16 H + 24 T. Would you challenge its fairness? • Yes • No
Suppose you thought a coin was fair and flipped it 300 times and it came up 160 H + 240 T. Would you challenge its fairness? • Yes • No
