If we can take two paths with different inclination angles and different length, do we know which one to choose? It depends on what we want to save: time or legs.

If you are in a hurry it's better to take velocity first on a steep slope and then quickly get to the goal. It takes much more time to gain the velocity on a straight road.

The shorter street does not always happen to be the quickest. Sometimes, when we hurry up for the train, it's worth taking the longest but steepest street.

Physicists call the track you can see in the picture 'the curve of the shortest time', i.e. a brachistochron. Mathematicians call it the circular curve, i.e. cycloid. For example, this is the curve plotted by the valve in your bicycle when you are riding. Such a curve is also plotted by this flashing yo-yo.

The task of finding the shortest fall was set by Swiss mathematician, Johann Bernouilli in 1696 and the problem was soon solved by his brother, Jacob Bernouilli, who was both physicist and mathematician. The so called Bernouilli effect explains how aircraft lifts in the air and why a small ball hovers in the stream of air from a hair-dryer.

The brachistochron case studies are all but banal. They led to the birth of a new branch of theoretical mechanics, so called variational calculus.