The Calculus Diaries - Jennifer Ouellette [109]
But let’s say our starting function gives us a bona fide curve with no straight lines or triangles to assist us; now things become complicated. We’ve already seen a method for approximating the area under a curve in chapter 1: the aptly named method of exhaustion pioneered by Eudoxus, whereby we fill in the curve with a series of rectangles for which it is a simple matter to determine the area. We calculate those individual areas, then add them all together to get an approximation of the area under the curve. The smaller the rectangles we use, the more of them it takes to fill the area under the curve, and the closer the approximation. We literally could do this forever, using infinitesimally small rectangles.
Luckily for us, there is another way: taking an integral. It’s a bit harder than finding the area of a triangle, but it simplifies matters greatly when trying to determine the area under a curve, so it’s worth walking through the process for v = at. (I will spare you the full derivation. You’re welcome.)
We can write out our question mathematically like this: h(t)=∫v(t)dt. This just says that we are integrating velocity (v) over time (t), adding all those instantaneous velocities together to determine our position (h).
Thanks to our handy velocity function, we know that v=at, so we can replace v(t) with at to get h(t ) = ∫at dt .
Another handy rule of calculus is that whenever you integrate a constant multiplied by a function, like at, you can bring that constant outside the integral symbol, like this:
h(t)=a∫tdt.
Now we can get rid of both our integral symbol and the dt by looking up the integral for t, which turns out to be We’ve picked up a constant because of another hard-and-fast rule of calculus: Whenever you take an indefinite integral—i.e., when no beginning and ending point is specified—your answer is going to have a constant (hence the waggish habit of physicists to jokingly add “plus a constant” to random observations). It makes sense if you think about it for a moment: The integral corresponds to the area under a curve, which by definition describes a given range. Even if we don’t know what that range is, we still need a placeholder in our equation: c represents that constant of unknown value.
We plug all of that into our equation to get this:
Look familiar? It’s our best buddy, the parabola! So now we know that if our velocity increases like a straight line (v=at), our position increases like a parabola (also written as And we can prove it mathematically.
Fun with Functions. But the point is, we’ve found our position function in just a few easy steps: Now we can determine our position (h) for any value of t.
Remember that in our free-fall scenario, b = 200 and a = −32. For instance, what is our position (h) when t = 1? It’s 184 feet. When t = 2, h = 136 feet, when t=3,h = 56 feet, and so on. In fact, we can devise an algebraic equation from our position function to determine when h will equal 0 and we will hit the ground if we fell from atop the Tower of Terror. Since we’re solving for t, it looks like this:
Plug those numbers into our formula like this: . The minus signs cancel out and we get
Now it’s just a matter of factoring down until we get . Our handy calculator tells us the square root of 2, we divide 5 by that, and the answer is t = 3.5. So we will hit the ground and go splat 3.5 seconds after we start falling.
APPENDIX 2
Calculus of the Living Dead
Time to nut up or shut up.
—TALLAHASSEE, Zombieland
A particularly virulent form of human-adapted mad cow disease sweeps across the United States in the 2009 hit film, Zombieland, transforming most of the nation’s