An optimal halfpipe is one that has the fastest line between 2 horizontal points only driven by gravity.
Here is a "half" of an optimal halfpipe - if you have the fastest way to the middle, the other half have to look the same!
I added a straight comparison curve and it is much slower!
It it a "calculus of variations"-problem, see Leonhard Euler und Joseph-Louis Lagrange.
But I calculate it with "Finite Elements" and a "Dynamic Programming" Idea, see Richard Ernest Bellman. I neglect rotation and rolling resistance as well as aerodynamic drag of the ball.
The size of model doesn't matter, it is always the same curve! I made also a little startbox to start the two iron ball simultaneous.
I made the model 25 cm, this is too long for my own Printer. I scaled it all down to 80 % (also the startbox)! I layed the model down with the straight course to the bottom and had support only on the side of the optimal curve.
Overview and Background
Optimal Line between two points, only driven by gravity
Lesson Plan and Activity
9th, 10th Grad:
Why is the staight curve slower?
Why is the shape of a "Real Halfpipe" not the same?
12th Grad, Higher Education:
Learn Finite Elements
- Learn Dynamic Programing
1 x halfpipe model
1 x startbox
2 x iron ball < 10 mm