Today is July 20. Remember when Olin Teague kept trying to make a national holiday of it?
Of course you don't. In fact, you're probably saying, “Who the heck was Olin Teague?”
That's okay. He's been gone awhile. U.S. Representative Olin Teague held a congressional seat from Texas for a big chunk of the 20th century. He chaired the Select Committee on Astronautics and Space Exploration. The Apollo moon program occurred on Teague's watch. You could say he was a big space booster.
By now you've recalled (or figured out) that July 20 is the anniversary of the first Apollo landing on the moon, the Apollo 11 mission that took Neil Armstrong and Buzz Aldrin down to its dusty surface. Sometimes, in a fit of nostalgia, I bother to mention it to my summer school students (if I'm teaching summer school that year, of course). Their reaction is usually dispiriting.
The last time I did it, I wrote “July 20” on the board and left it there for a while. Since I don't normally bother writing the day's date on the board, it was anomalous behavior. Successfully stifling any curiosity—intellectual or otherwise—my calculus students did not ask the date's significance.
So I did.
“Does anyone know what today it?”
A couple of my more insightful students said, “July twentieth!”
I gave them a sickly smile.
“Thank you for the obvious answer. Can anyone dig a little deeper than that?”
“It's my niece's birthday!”
“Okay,” I said, with great forbearance. “I hope you got your niece an appropriate birthday present.”
“I don't know. I'll find out this afternoon when Mom gets back from shopping.”
Things were not developing in an encouraging direction. I abandoned subtlety.
“Today is the anniversary of the first moon landing,” I said.
They looked at me, mostly blankly. A couple of students said “oh” without much enthusiasm in their voices.
“Yes, I realize we're talking about ancient and boring history,” I said. “No doubt you've heard of the Apollo program and seen photos or videos of it, but it's probably difficult to imagine how exciting it was to see it all occur in real time. We saw the lift-offs and moon landings on live television, just as it occurred. It was a thrilling time to be a witness to history.”
“My father said he actually got hives on the day of the moon landing,” said one student, a small smile on her face. “But he's an electrical engineer.”
“Oh, an engineer. That explains it, of course,” I replied.
Students nodded their heads as if everyone knew that's the way engineers are. They're easily susceptible to hives and other allergy-related afflictions. It's possible—or even likely—that they were dutifully humoring me, having no particular opinions of their own. One should always encourage a teacher who is off-topic and telling stories instead of lecturing.
“The space program was a great motivator when I was in high school. It was exciting to all of us who were interested in math, science, and engineering. The Apollo moon rocket generated seven and a half million pounds of thrust to lift a six million pound vehicle from the launch pad. It began painfully slowly, but the final stage reached speeds of seven miles per second as it left earth orbit. That's about twenty-five thousand miles per hour. Residents of Hawaii actually got to see the third stage of Apollo 11 light up overhead as it pushed the spacecraft toward the moon.”
Some of my students goggled as I reeled off the numbers. I had either captured their attention or they were good at feigning interest. (Either—or both—is entirely possible.) I turned to the board and wrote “F = ma.”
“What does that mean?”
Many students were ready to blurt out the answer.
“Force equals mass times acceleration.”
I paused for a long moment, waiting.
“The mass wasn't constant!” a student proudly announced.
“Exactly!” I said. “The first stage was burning off fuel at a rate of fifteen tons per second.”
Even some of the more jaded students were a bit slack-jawed with wonder now.
“So, what to do? Well, I broke the time interval into short segments.” On the board I wrote t0, t1, t2, and so on. “Since I knew the rate of fuel consumption, I knew the mass of the rocket at the beginning of each subinterval. I could compute the acceleration for that interval and calculate the rocket's change in altitude. The computations were simple, but there were a lot of them. How could I get my results to be even more accurate?”
“Do even more calculations,” they said. “Use shorter subintervals!”
“Quite right! And how does that compare to something you've learned in this class?”
“Smaller intervals for more accuracy in Simpson's rule. Or in Riemann sums!”
“Indeed,” I agreed.
“Hey, you were giving a math lecture after all!” cried a student.
“Gotcha!” I admitted.
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8 comments:
Hey, it sounds like for once those students of yours actually got something for once. ;) Anyway, it seems like they're doing better than that group of South American students that Richard Feynman talked to once who couldn't related a formula they'd been taught to an actual physical phenomenon.
So...did you get them all the way to the limit? Was there... calculus?
Dammit: I had to do it myself:
Assume that the rocket is designed to supply constant force. Thus,
a(t)=(M-alpha*t)/f, where M is the initial mass of the rocket, and alpha is the rate at which mass is depleted. a(t), of course, is acceleration at time t.
Then, since velocity is the anti-derivative of acceleration, we hold that:
v(t)=Int(o,t){a(x)dx}
Thus, v(t)= t*((M/f)-(alpha/(2*f))*t).
Note that this is only valid for the interval in which the first stange burns off, and depends on M>alpha (you know, because the rocket weighs more than the fuel it burns).
Worse, I inverted the acceleration function, so that's all wrong.
To make things even more fun, I initially tried these calculations before I knew calculus. I told my students that the initial moon landing occurred the summer before I enrolled in Calculus I, but I was able to cobble together an approximation of the Apollo-Saturn's initial ascent by incrementally computing the mass, acceleration, and position of the vehicle. My students were impressed in a couple of ways, one of them involving shaking their heads in sympathy for my lost and squandered youth (I'm sure).
Of course, the computations pretty much ignored air resistance, the tilt of the rocket after it left the pad, and the diminution of gravitational acceleration at higher altitudes (rather negligible in the current instance).
Oops, you're right, AnyEdge! Want to take another stab at it? I can delete your initial attempt, if you like, lest it lead the innocent astray.
The correct answer is:
V(t) = (f/alpha)*(lnM - ln(|-alpha*t+M|))
Valid so long as M>alpha. Or we learn to build rockets with negative masses.
God forbid NASA is trolling the blogosphere for grade school level calculus to fly their rockets. I don't think we need fear anyone being led astray.
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