I recognize the lion by his paw

One of my co-workers is very involved with the Boy Scouts (an organization I used to support financially until they decided being gay meant you couldn’t keep your hands off of young boys, but that’s another point altogether). We were chatting a few days ago and he mentioned a competition called the ‘Pinewood Derby’ which has to be one of the coolest competitions I have ever heard of! Each child is given a block of wood (made of pine) with two notches for wheels, four plastic wheels and four nails. The finished car must use all nine pieces, must not exceed a certain weight (usually five ounces), must not exceed a certain length and must fit on the track used by that particular scout pack. On competition day, all the cars are placed at the top of the track and released at the same time – first car to the bottom wins[1].

I started thinking about how I would build my own Pinewood Derby car and it made me think of one of my favorite physics stories – the story of the Brachistochrone. Here is the story for those of you who aren’t dorky engineers and then for those of you who are, I have a question.

The word Brachistochrone comes from the Greek words brachistos (shortest) and chronos (time)[2]. In 1696, Johann Bernoulli (tutor to Euler and l'Hôpital and father of Daniel Bernoulli for whom Bernoulli’s Principle is named) posed the following problem having already solved it himself:

Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.

Clearly, the shortest path would be a straight line, but would that be the quickest? As it turns out, no...the quickest path between these points is the upside-down cycloid (path drawn by a point on the rim of a rolling wheel) which is vertical at point A and intercepts point B as shown here:

The three signed solutions sent to Johann Bernoulli represent a who’s who of late seventeenth century mathematics – Johann’s older brother Jacob Bernoulli, Gottfried Leibniz and Guillaume de l'Hôpital. A fourth, unsigned solution also arrived at Benoulli’s doorstep. Upon recognizing it as the work of Isaac Newton, Bernoulli is said to have exclaimed I recognize the lion by his paw. (Interestingly, the cycloid is also the tautochrone – the curve along which a marble placed at any point will reach the bottom at the same time)

This got me thinking – and here is the question for my engineering colleagues – wouldn’t the fastest Pinewood Derby car be one in which the c.m. followed (to as great a degree as possible) the brachistochrone curve? Imagine hollowing out the piece of wood, standing it upright and placing a metal sphere attached to a spring on the inside. By choosing a spring with the right gain, one could make the c.m. of the system follow a [nearly] cycloidal trajectory during the descent. I’m guessing this is well outside the rules, but I’m also guessing that it would look pretty neat to the observer – one car speeding ahead of the others despite being rather odd looking and having no regard for air-resistance :-)