Brachistochrone and Tautochrone curves
Calculus
Which ramp shape gets a skier to the bottom fastest? Use a real dry-ski jump as your case study.
Where this idea comes from
Start here — this is the source that inspired this exploration.
Introduction
A dry-ski freestyle park like the one in Leysin, Switzerland, sends skiers down a ramp from a height of around 26 metres before they leave the ground. That ramp could be built straight, curved like part of a circle, or curved like a cycloid — the shape that gets an object to the bottom faster than any other, known as the brachistochrone. Compare descent times for different ramp shapes and find out why the cycloid wins.
Guiding Questions
- Which ramp shape gets you to the bottom fastest — straight, circular, or something else? Make a prediction before you calculate anything.
- Compare descent times for two or three candidate shapes using energy arguments and numerical calculation.
- The brachistochrone (a cycloid) beats every other shape. Using the real 26-metre ramp height, how much faster is it than a straight line?
- The same curve is also tautochrone: a ball released from any point on it reaches the bottom in the same time. Check that with a calculation or a simple experiment.