You design a roller coaster that has an initial drop from some height "h" followed by a loop and then the ride comes to a stop. If the loop of a circle with a diameter of 50 m, what is the minimum height you would need to make sure nobody falls out? (Ignore friction)

Question

You design a roller coaster that has an initial drop from some height "h" followed by a loop and then the ride comes to a stop.  If the loop of a circle with a diameter of 50 m, what is the minimum height you would need to make sure nobody falls out? (Ignore friction)

drwls · Answer

Radius R = 25 m

V^2/R at the top must equal or exceed g.

For the minimum V, set V^2/R = g
V = sqrt(Rg) = 15.6 m/s

h must be located far enough above the top of the loop so that
(1/2)MV^2 = M g h

h = V^2/(2g) = gR/2g = R/2

Total height above bottom of the loop = h + 2R = 2R + R/2 = (5/2) R