What is the fastest safe speed the roller coaster can go over the top of the hill?

Question

A roller coaster is moving over the top of a circular hill that has a diameter of 64m. What is the fastest safe speed that the roller coaster can go over the top of the hill? ( that is, at what speed will the cart begin to lose contact with the track?) assume a coefficient of friction of 0.6

Answer 1

At the top of the hill, the net downward force must equal the centripetal force. The net downward force is the weight minus the upward normal force. But at the maximum safe speed, the upward normal force is zero, and the vehicle is on the border between flying off the hill and rolling down the hill. That leaves the weight exactly equal to the centripetal force.

mg = mv_max²/(d/2),

where d is the diameter of the circular hill.

g = 2v_max²/d

v_max = √(gd/2)

Plug in

g = 9.8 m/s²

d = 64 m

and get v_max in m/s.