It skims over pyrolysis; chemiluminescence, where the chemical reaction (combustion/oxidation) produces excited atoms and molecules that need spit out (emit) blue light to get to their ground state); and the incandescent light emission of microscopic soot particles which produce the yellow parts of the flame.
I’m not sure who the guy chained to the rock is. It might be Prometheus, who stole fire from the gods, but I don’t remember him being sent into hell in the myth.
The bright blue iridescence of the wings of this insect results from the way light refracts through the thin layered membranes of the wings.
When you look at the sunlight reflected off this black insect’s wings at just the right angle, they blaze bright blue. The phenomena is called iridescence, and results from the way different wavelengths of light refract through the wing membrane. Blue light is of just the right wavelength that the light reflected off the top of the membrane and the light that’s refracted through the membrane constructively interfere. The Natural Photonics program at the University of Exeter has an excellent page detailing the physics of iridescence in butterflies (Lepidoptera), and the history of the study of the subject.
Surface tension supports the water strider on the surface of the still water of the creek.
Down at the creek the water striders are out. They can stand, walk and jump on the surface of the water without penetrating the surface because of the force of surface tension that causes water molecules to stick together — it’s the same cohesive force that make water droplets stick to your skin. I got a decent set of photos to illustrate surface tension.
The water striders still create ripples on the surface of the water, even though they never break the surface.
The green canopy that over hangs the creek allows for some nice photographs.
One of the middle-school projects is to build a little solar water heater. By simply pumping water through a black tube that’s sitting in the sun, you can raise the temperature of the water by about 15°C in about 15 minutes.
The solar water heater in action.
Next year I want to try building an actual solar water heater, similar to the passive air heater my students built two years ago, with the tubing in a greenhouse box to see just how efficient we can make it.
The area around the Sun is blue on Mars because the gasses in the thin atmosphere don't scatter much, but the Martian dust does (it scatters the red). Image via NASA.
The dust in Mars’ atmosphere scatters red, while the major gasses in Earth’s atmosphere (Nitrogen and Oxygen) scatter blue light. Longer wavelengths of light, like red, will bounce off (scatter) larger particles like dust, while shorter wavelengths, like blue light, will bounce of smaller particles, like the molecules of gas in the atmosphere. The phenomena is called Rayleigh scattering, and is different from the mechanism where different molecules absorb different wavelengths of light.
Ezra Block and Robert Krulwich go into details on NPR.
Blue sky in the upper right, but the dust scatters the red light.
The discussion started off with projectiles. If you’re floating in space — zero gravity — and throw something, like a basketball, away from you, you’ll be pushed off in the opposite direction. In fact, if you throw something that has the exact same mass as you do away from yourself, you’ll move off in the opposite direction with the exact same speed as the thing you threw.
Then I brought up rockets, and how they’re expelling gas to move them forward. I think it was the phrase, “expelling gas” that did it. The next question, which the student brought up somewhat circumspectly, sidling around the issue and the language, was (more or less), “So if you expel gas in space will you move off in the other direction?”
The simple answer, appropriate to that stage of the discussion, was, of course, “Yes.”
Which lead to to, “What about spitting?”
“Yes.”
“What about, you know, peeing?”
“Yes, except …”
At that point I thought it would be wise to rein it in a little, and make a further point about the whole action-reaction thing.
“You see, if you expel anything, wouldn’t it just be stuck in your spacesuit with you? Then you’re not really expelling it, it’s still attached to you, so you wouldn’t really move. What would be more useful would be to collect it in something like a spray can or a squeeze bottle. Then you can just squirt it out opposite the direction you want to go in to control your movement.”
This produced a moment of thoughtful silence as they figured out the logistics.
Notes
I thought this was a useful conversation to have. The students were interested and animated. And I believe it’ll be memorable too.
An artist's concept depicts the Deep Space 1 probe with its ion engine operating at full thrust. Image via NASA.
P.S.: I’d wanted to talk about ion drives, which operate on the same reaction principle, but are much cooler (after all they’re used in Star Trek). Instead of burning fuel to create the propulsive force ion drives generate an electric field that ejects charged particles; we’d been talking about ions and charged particles earlier in the day. However, I decided on the day that it would just complicate what was a new issue. I’ll probably bring it up this week though as we recurse through Newton’s Laws.
PhD Comics does a wonderful job of explaining of sub-atomic particles: what we know, what we don’t know. What’s particularly great about this video is that it goes into how physicists are using the Large Hadron Collider to try to discover new particles: by making graphs of millions of collisions of particles and looking for the tiniest of differences between different predictions of what might be there.
I also like how clear they make the fact that science is a processes of discovery, and what fascinates scientists is the unknown. Students do experiments all the time and if they don’t find what they expect — if it “doesn’t work” — they’re usually very disappointed. I try my best to let them know that this is really what science is about. When your experiment does not do what you want, and you’re confident you designed it right, then the real excitement, the new discoveries, begin.
Yesterday we used calculus to find the equation for the height of water in a large plastic water bottle as the water drained out of a small hole in the bottom.
Perhaps the most crucial point in the procedure was fitting a curve to the measured reduction of the water’s outflow rate over time. Yesterday, in our initial attempt, we used a straight line for the curve, which produced a very good fit.
Figure 1. The change in the outflow rate over time can be well approximated by a straight line.
The R2 value is a measure of how good a fit the data is to the trendline. The straight line gives an R2 value of 0.9854, which is very close to a perfect fit of 1.0 (the lowest R2 can go is 0.0).
The resulting equation, written in terms of the outflow rate (dV/dt) and time (t), was:
However, if you look carefully at the graph in Figure 1, the last few data points suggest that the outflow does not just linearly decrease to zero, but approaches zero asymptotically. As a result, a different type of curve might be a better trendline.
Types of Equations
So my calculus students and I, with a little help from the pre-Calculus class, tried to figure out what types of curves might work. There are quite a few, but we settled for looking at three: a logarithmic function, a reciprocal function, and a square root function. These are shown in Figure 2.
Figure 2. Example curves that might better describe the relationship between outflow and time.
I steered them toward the square root function because then we’d end up with something akin to Torricelli’s Law (which can be derived from the physics). A basic square root function for outflow would look something like this:
the a coefficient stretches the equation out, while the b coefficient moves the curve up and down.
Fitting the Curve
Having decided on a square-root type function, the next problem was trying to find the actual equation. Previously, we used Excel to find the best fit straight line. However, while Excel can fit log, exponential and power curves, there’s no option for fitting a square-root function to a graph.
To get around this we linearized the square-root function. The equation, after all, looks a lot like the equation of a straight line already, the only difference is the square root of t, so let’s substitute in:
to get:
Now we can get Excel to fit a straight line to our data, but we have to plot the square-root of time versus temperature instead of the just time versus temperature. So we take the square root of all of our time measurements:
Time
Square root of time
Outflow rate
t (s)
t1/2 = x (s1/2)
dV/dt (ml/s)
0.0
0
3.91
45.5
6.75
3.52
97.8
9.89
2.94
140.9
11.87
3.52
197
14.04
3.21
257
16.05
3.01
315.1
17.75
2.81
380.1
19.50
2.53
452.9
21.28
2.23
529.6
23.01
1.92
620.7
24.91
1.69
742.7
27.25
1.45
We can now plot the outflow rate versus the square root of time (Figure 3).
Figure 3. Linear trend relating the outflow rate to the square root of time. The regression coefficient (R2) of 0.9948 is better than the simply linear trend of outlfow rate versus time (which was 0.9854).
The equation Excel gives (Figure 3), is:
and we can substitute back in for x=t1/2 to get:
Getting back to the Equation for Height
Now we can do the same procedure we did before to find the equation for height.
First we substitute in V=πr2h:
Factor out the πr2 and move it to the other side of the equation to solve for the rate of change of height:
Then integrate to find h(t) (remember) :
gives:
which might look a bit ugly, but that’s only because I haven’t simplified the fractions. Since the radius (r) is 7.5 cm:
Finally we substitute in the initial value (t=0, h=11) to solve for the coefficient:
giving the equation:
Plotting the equations shows that it matches the measured data fairly well, although not quite as well as when we used the previous linear function for outflow.
Figure 4. Integrating a square root function for the outflow rate gives a modeled function for the changing height over time that slightly undermatches the measured heights.
Discussion
I’m not sure why the square root function for outflow does not give as good a match of the measurements of height as does the linear function, especially since the former better matches the data (it has a better R2 value).
It could be because of the error in the measurements; the gradations on the water bottle were drawn by hand with a sharpie so the error in the height measurements there alone was probably on the order of 2-3 mm. The measurement of the outflow volume in the beaker was also probably off by about 5%.
I suspect, however, that the relatively short time for the experiment (about 15 minutes) may have a large role in determining which model fit better. If we’d run the experiment for longer, so students could measure the long tail as the water height in the bottle got close to the outlet level and the outflow rate really slowed down, then we’d have found a much better match using the square-root function. The linear match of the outflow data produces a quadratic equation when you integrate it. Quadratic equations will drop to a minimum and then rise again, unlike the square-root function which will just continue to sink.
Conclusions
The linearization of the square-root function worked very nicely. It was a great mathematical example even if it did not produce the better result, it was still close enough to be worth it.
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