My theme is a rough-and-ready back-of-the-envelope method used so automatically that we often overlook it, yet used so often that it needs a name. I call it balancing.

A typical application: What is the Earth's approximate orbital speed v? With mass m, the Earth has kinetic energy ∼ mv2 (ignoring the factor of one-half thanks to the magic of the twiddle1). At an orbital radius r, it has potential energy ∼ GMsunm/r (hiding the minus sign in the twiddle). Balancing these energies—equating them up to a twiddle—gives mv2GMsunm/r and

vGMsunr,
(1)

which, at r = 1 AU, is 30 km/s.

A subtler application: As the radius R of an atomic blast grows in time t since the explosion, how does R depend on t and on...

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