Path that requires the least time to travel along

[bbal]

Homework Statement

We have slope, over which there's a point (P). The point is connected to the slope with a straight line. Find the line, a small ball would travel along the fastest.

Relevant Equations

S=(at^2)/2

 

[DaveC426913]

Show your work so far.

 

[bbal]

Show your work so far.

It's basically all you can see on the picture. I took it as a starting, that if we connect the "top" of a circle to any other point of the circle with a straight line, the time to travel along each would be the same. Then I tried to sketch a circle, such that, point P is on "top" of it and the slope is tangent to it.

 

[etotheipi]

Well funnily enough you actually drew the correct trajectory on your left hand diagram, but can you justify it?

The obvious, but not particularly elegant, approach, is to consider an arbitrary trajectory at some angle $\alpha$ to the downward vertical through the point P, find the component of gravitational acceleration parallel to the rail, find the length of this rail, and vary $\alpha$ in such a way to minimise the time.

The answer might suggest a simpler line of reasoning! If you remember your circle theorems...