Thursday, August 12, 2010

Jules Verne Likes Math For Its Own Sake

Jules Verne liked his mathematics. While describing Dr. Ferguson, the rugged, extremely well traveled protagonist of Five Weeks In A Balloon, Verne posits the following question:
"Given the number of miles traveled by the doctor in making the circuit of the Globe, how many more had his head described than his feet, by reason of the different lengths of the radii? - or, the number of miles traversed by the doctor's head and feet respectively being given, required the exact height of the gentleman?
To start, let's draw a diagram:

Figure 1: Dr. Ferguson travels the world.

The blue and green thing is the world, and the gray guy in a suit is Dr. Ferguson. As he travels (we'll assume at sea level) across the world (which we're assuming is spherical, with a radius r), he creates an angle θ between his starting point, the center of the earth, and his ending point. He thus moves across an arc length d. However, because of Ferguson's height (h), the distance between Ferguson's head and the center of the earth is greater than the distance between his feet and the center of the earth, so his head move across a different arc length, dhead. We're trying to solve for Δd, or dhead - d

From here, it's simply geometry. We know that d = rθ, so θ = d/r. Similarly, we know that dhead = (r + h)θ. By substitution, we determine that dhead = (r + h)(d/r). Let's simply...
dhead = (r + h)(d/r)
dhead = dr/r + dh/r
dhead = d + dh/r
Since  Δd = dhead - d, we can substitute and get Δd = (d + dh/r) - d = dh/r.

The difference in distance traveled by Ferguson's head than that traveled by his feet is thus the distance his travels across the earth times his height, divided by the radius of the earth. While we have to rely on a bunch of assumption to get this simply answer, it's likely a very close approximation.

Now, to finally answer Verne's question: Assuming Dr. Ferguson is 6 feet tall, and that he has traveled around the world exactly one time, we can compute that Δd = (circumference of the earth)(6 feet)/(radius of the earth), or that Δd = 2π(6 feet) = 12π feet= 37.699 feet. Which makes sense, as the earth's radius is huge compared to the height of anyone who isn't Yao Ming. (And, comparatively speaking, the earth's radius is still pretty big for him too.)

If you cared enough to read all that, I'm sure Jules Verne would be extremely happy with you. At any rate, I'm very happy with you.

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