Question

A car in an amusement park travels without friction on a vertical
loop of 20 m radius. Calculate from what minimum height h it needs to be
released from rest in order to remain on the track at the top of the loop.

Answer 1

It has to get 40 m high
when it gets 40 m high, it must have a centripetal acceleration of 1 g or it no longer touches its track.
so
Ac = v^2/r = 9.81 m/s^2

v^2 = 20*9.81
v = 14 m/s

now it has to drop a distance
m g h = (1/2) m v^2 to the TOP of the wheel

h - 40 = .5 v^2/g
but we know v^2/g = 20
so
h- 40 =10 meters
so height h = 50 meters
yikes!