proj

Using conservation mechanical energy between points 𝐴 and 𝐵:

(1)

The velocity at points B and C are equal and can found from equation (1):

(2)

The acceleration between points 𝐴 and 𝐵 is the same as the acceleration between points 𝐶 and 𝐷. This acceleration is provided by gravity and depends on angle 𝜃 and is given by

𝑎 = 𝑔 sin 𝜃 (3)

The traveling time from points A to B and from points C to D is given by

(4)

Where

(5)

The motion on the horizontal surface from B to C is uniform (without acceleration) and is given by

(6)

The total time is given by

(7)

In this project you will find the minimum time for a particular trapezoidal path between two given points A and B. In general the path of the shortest time between two give points in the presence of gravity is given by brachistochrone (cycloid) curve. Write a paragraph about brachistochrone (cycloid) curve.