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.
2. Some of your worst days lie ahead. Graduation is a happy day. But my job is to tell you that if you are going to do anything worthwhile, you will face periods of grinding self-doubt and failure.
Charles Wheelan provides an excellent perspective on what should be important in a commencement address.
I particularly like this warning about the danger of working only for rewards:
8. Don’t model your life after a circus animal. Performing animals do tricks because their trainers throw them peanuts or small fish for doing so. You should aspire to do better.
And this point on conservation and the real meaning of being conservative:
3. Don’t make the world worse. I know that I’m supposed to tell you to aspire to great things. But I’m going to lower the bar here: Just don’t use your prodigious talents to mess things up. Too many smart people are doing that already.
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.
We successfully harvested and processed three chickens during last week’s interim. It was my first time going through the entire process, but fortunately we had a very experienced guide in Dr. Samsone who also happens to be a vet.
The interim focused on where food comes from (students also saw the documentary “King Corn”), and the cleaning of the chickens was tied into our Biology students’ study of anatomy (I’d done fish and squid before). Unfortunately, I was unable to find someone who knew how to read the entrails so we could tie the process into history and language arts as well.
Student holds a kidney. A heart is in the background.
When we were done with the processing and analysis, Mr. Elder cooked the chickens on our brand new grill (which worked quite well he says). The chickens were free-range (donated by Ms. Eisenberger), but a little on the old side, at about 7 months old; the chickens you buy at the grocery are somewhere around 2 months old.
Dr. Samsone recommended that next time we raise the chickens ourselves from chicks, which I’d love to try, but I suspect would run into some serious resistance from the students. We’d only had the chickens we harvested for five minutes before they’d all been given names. Raising chickens from chicks would bring a whole new level of anthropomorphizing.
Chicken on the grill. The culmination of the interim.
References
Being new to the chickens, I spent a bit of time researching how it is done.
Ken Bolte, from the Franklin County Extension of the University of Missouri, recommended the University of Minnesota’s Extension site on Home Processing of Poultry (the page on evisceration provided an excellent guide), as well as Oklahoma State’s much briefer guide (pdf).
Dr. Samsone recommended the series of videos from the Featherman Equipment Company. Videos are particularly useful for novices like myself.
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.
In addition to eating the bulbs of the radishes, the leaves are also edible. I heartily endorse Clotilde Dusoulier’s Radish Leaf Pesto. The slight spiciness of the leaves gives it a delightful frisson.
Radish leaf pesto.
Pesto recipes are pretty flexible. I added some fresh cilantro from the garden, some frozen basil leaves, used ground almonds for the nut component, a bit of Manchego for the cheese, and doubled the garlic. I also added a little white wine to reduce the viscosity. I quite liked the end result — we had it on pasta — even if some others though it was a little too adventurous.
I punched a small hole (about 1mm radius) in a one gallon plastic bottle and had my students measure the rate at which water drained. Even though the apparatus and measurement technique was fairly rough, we were able to, with a little calculus, determine the equation for the height of the water in the bottle as a function of time.
Figure 1. A student uses a stopwatch to measure the outflow rate of water from the plastic bottle.
Introduction
Questions about water draining from a tank are a pretty common in calculus textbooks, but there is a significant difference between seeing the problem written down, and having to figure it out from a physical example. The latter is much more challenging because it does not presuppose any relationships for the change in the height of water with time; students must determine the relationship from the data they collect.
The experimental approach mimics the challenges faced by scientists such as Henry Darcy who first determined the formulas for groundwater flow (Darcy, 1733) almost 300 years ago, not long after the development of modern calculus by Newton and Leibniz (O’Conner and Robertson, 1996).
Procedure
Figure 2. The apparatus.
I punched a small hole, about 1mm in radius, in the base of a plastic, one-gallon bottle. I chose this particular type of bottle because the bulk of it was cylindrical in shape.
Students were instructed to figure out how the rate at which water flowed out (outflow rate) changed with time, and how the height of the water (h) in the bottle changed with time. These relationships would allow me to predict the outflow rate at any time, as well as how much water was left in the container at any time.
Data collection:
To measure height versus time, we marked the side of the bottle (within the cylindrical region) in one centimeter increments and recorded the time it took for the water level to drop from one mark to the next. There were a total of 11 marks.
To measure the outflow rate, we intercepted the outflow using a 25 ml beaker (not shown in Figure 2), and measured the time it took to fill to the 25 ml mark.
Results
Time elapsed since last measurement
Height of water in container
Time to fill 25ml beaker
Δt (s)
h (cm)
tf (s)
0.0
11
6.4
45.5
10
7.1
52.3
9
8.5
43.1
8
7.1
56.1
7
7.8
60.6
6
8.3
57.6
5
8.9
65.0
4
9.9
72.8
3
11.2
76.7
2
13.0
91.1
1
14.8
122.0
0
17.3
Table 1: Outflow rate, water height change with time.
To analyze the data, we calculated the total time (the cumulative sum of the elapsed time since the previous measurement) and the outflow rate. The outflow rate is the change in volume with time:
So our data table becomes:
Time
Height of water in container
Outflow rate
t (s)
h (cm)
dV/dt (ml/s)
0.0
11
3.91
45.5
10
3.52
97.8
9
2.94
140.9
8
3.52
197
7
3.21
257
6
3.01
315.1
5
2.81
380.1
4
2.53
452.9
3
2.23
529.6
2
1.92
620.7
1
1.69
742.7
0
1.45
Table 2. Height of water and outflow rate of the bottle.
The graph of the height of the water with time shows a curve, although it is difficult to determine precisely what type of curve. My students started by trying to fit a quadratic equation to it, which should work as we’ll see in a minute.
Figure 3. The decrease in height with time is not a straight line (is non-linear).
The plot of the outflow rate versus time, however, shows a pretty good linear trend. (Note that we do not use the first three datapoints (in Table 2), which we believe are erroneous because we were still sorting out the measuring method.)
Although I’ll note here that the data should ideally be modeled using a square root equation (Torricelli’s Law), that is beyond the present scope of the problem (we’ll try that tomorrow as a follow exercise).
Plotting the data in Excel we could add a linear trendline.
Figure 4. The outflow rate decreases linearly with time.
For the linear trendline, Excel gives the equation:
The y-axis is outflow rate (the change in volume with time), and the x axis is time (t), so the linear equation becomes:
Notice that this is a differential equation.
To determine the rate of at which the height of water in the container is changing, we need to recognize that the container is cylindrical in shape, and the volume (V) of a cylinder depends on its radius (r) and height (h):
which can be substituted into differential in the rate equation:
since π and the radius (r) are constants (since the jug’s shape is a cylinder), they can be pulled out of the differential:
Dividing through by πr2 solves for the rate of change of height:
Isolating the coefficients gives:
This equation should give the rate at which the height changes with time, however, if you look at it carefully you’ll realize that for the time range we’re using (less than 800 seconds) the value of dh/dt will always be positive. We correct this by recognizing that the outflow rate is a loss of water, so a positive outflow should result in a negative change in height, therefore we rewrite the equation as:
which gives:
Now comes the calculus
Now, we can use this rate equation to find the equation for height versus time by integrating with respect to time:
to get:
And all we have to do to the find the constant of integration is substitute in a known point, an initial value. As is often the case, the best point to use is the starting point where t=0 makes the rest of the calculations easier. In our case, when t=0, h=11:
so:
And our final equation becomes:
which is a quadratic equation as my students guessed before we did the calculus.
Does it work?
You will notice that in the math above, we never use the height data in determining equation for height versus time; all the calculations are based on the trendline for the outflow rate versus time.
As a result, we can compare the results of our equation to the actual measurements to see if our calculations are even close. Remarkably, they are.
Figure 5. The results from our equation (modeled) match the measured heights so well, the data points are difficult to distinguish on the graph.
Discussion
Despite all the potential for error, particularly, our relatively crude measurement techniques, and the imperfect cylindrical shape of our plastic bottle, the experiment went remarkably well.
Students found it quite challenging, and required some assistance even though this is a problem they have seen before in their textbook.
The problems in the textbook use Torricelli’s Law, which should much better describe the draining of a tank than the linear equation we find for dV/dt.
Torricelli’s Law:
where a is the area of the outlet hole, and g is the acceleration due to gravity.
In our actual experiment it is difficult to tell that a square root function would work better. Excel does not have an option for matching a square root function, so the calculations would become more involved (although it could be set up using Excel’s iterative solver or Goal Seek function).
Conclusion
Our experiment to use calculus to determine the rate of change of the height of water in a leaking plastic water bottle was a successful exercise even though the roughness of the data collection did not permit identification of the square root law for leakage.
The radishes did well this year. Planted in containers on March 29th (in St. Louis, USA), they were harvested one month later. The short, early season means that they’re a workable crop for school. Students can plant, harvest, and consume them all within a semester.
The CDC’s Fruit and Vegetable of the Month website has a little history, some information about the varieties, nutritional information, recipes, and more information about radishes. The University of Illinois Extension also has information about planting and growing.
NutritionData.self.com has some very nice graphical representations of the nutritional value of the food (although their serving size is 1 cup of slices, which seems a bit much).
These sites, however, focus on the radish bulbs, and not on the fact that the leaves are edible. Radish Leaf Pesto is quite good.
Harvested radishes. Both the red bulbs and the green leaves are edible. You'll note that radishes also spot a long tap-root.
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