Pinewood Derby Physics
Have you ever seen a pinewood derby race, a race that is held every year in many Cub Scout dens? I have worked with my grandson to enter one a couple of years, and being the engineer that I am, I wanted to use my knowledge of math and physics to perhaps win the race.
We tried to add weights to the car, decreasing the friction of the wheels (it is illegal to use ball bearings), and experimented with the shape of the pinewood derby car to give it a more aerodynamic shape. All of these things help to one degree or another, and all are well known in the Cub Scout community as a simple search will prove (https://www.wired.com/2010/12/pinewood-derby-physics/).
Tautochrone Curve (or Brachistrochrone Curve)
There is something that can affect the speed of an object down a pinewood derby track and that is the shape of the track. You would think that the shortest distance would give you the fastest time (We all know that the shortest distance between two points is a straight line). But in the real world with gravity pushing an object down a slope, the distance takes a peculiar shape, called the Tautochrone curve (also brachistrochrone curve), shown in the graphic above. The next simulation shows this.
Image courtesy of Curiosa Mathematica
Another strange property of the curve is that regardless of where an object is placed on the curve, the time required to reach the end is the same. The first illustration shows this, and is totally unintuitive.
The curve is a form of a cycloid (see https://en.wikipedia.org/wiki/Cycloid), that is the path a point on a wheel makes when it rolls on the ground. There is a lot of math involved in the description (but not too much for a student familiar with trigonometry).
The point to all this is that math is fascinating and has a lot of amazing applications for the student with an inquisitive mind. There is a lot of history behind the study of the cycloid, as noted from the wikipedia article cited above shows, so there is an endless supply of interesting applications of math and a lot of unsolved math mysteries for those who want to make a difference.