My old calculus book gives a formula for the curvature of a parametric arc in the plane — that is, an arc defined by two functions (x(t),y(t)) of one variable. For thirty years I didn’t think about the derivation of that formula. Just now it hit me (and I did the algebra to confirm) that, in terms of the complex plane (z=x+iy), the curvature formula is equivalent to
Im(z″/z′) / |z′|
This should improve my cubic approximations to transcendental curves.
To throw for maximum distance (on an infinite plane in a vacuum), you aim at 45° elevation; in other words, split kinetic energy evenly between vertical and lateral velocity. (I dimly remember having proven that, but am not awake enough to do it again now.)
It follows that the kinetic energy of a perfect vertical throw is twice the vertical kinetic energy of a perfect distance throw.
The arc of the latter is a parabola, of course, and its height is easily found to be a quarter of its length. Double that, because with double the energy you get double the altitude; and the result is consistent with Munroe’s estimate.
What about air resistance? The vertical throw has a moment of zero speed, but the distance throw’s minimum speed is 1/√2 the maximum; so it seems to me that the vertical throw suffers less air resistance (not even considering the thinner air up there), and therefore the altitude estimate given above is low.
A new father asked:
Are you charmed or annoyed to learn that your 3d geometric object is making a great toy for [baby, 8 mo.]?
I replied:
I have cats, so I’m not a bit surprised!
On a related note, if I had any knowledge of Web programming I’d make something analogous to this.