Quickest Way Down
The shortest distance between two points is a straight line, but thanks to gravity and acceleration, it is not the quickest way down.
The curve is called a brachistochrone, which means “shortest time”. This curve is quicker because it starts off steeply which allows the ball to accelerate quickly.
This curve is called a tautochrone, which means “same time”.
A tautochrone can start at any point on the cycloid, but always ends horizontally. A brachistochrone can end at any point on the cycloid, but always starts at the top of the cusp.
In 1696, famous Swiss mathematician, Johann Bernoulli set a challenge for other mathematicians of the time. The challenge was to find the curve that would give the quickest way down. Along with Johann Bernoulli, five other mathematicians submitted answers, they were Jakob Bernoulli (Johann’s brother), Isaac Newton, Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l’Hôpital.
The mathematicians all solved the problems in different ways, but all got the same answer, with some noting that the curve was part of a cycloid, and related to the tautochrone that had been discovered 40 years earlier.
This problem led to new mathematical tools for solving problems, which are still used today to solve problems involving the quickest time, or shortest distance.
Maths at Home
The curve used in this activity is called a cycloid. A cycloid is the curve drawn by a point of the rim of a wheel as it rolls along the ground. If we roll a circle inside another circle we get spirals. These are the same kinds of patterns made by a Spirograph. Try this online Spirograph (called Inspirograph) and see what kind of patterns you can make,
Can you make a cycloid? What do you need to make a cycloid?