Please help me with a physic problem about roller coaster?

Question

A frictionless roller coaster is given an initial velocity of V(intial) at height h= 26 m. The radius of curvature of the track at point A is R=38 m. The acceleration of gravity is 9.8 m/s^2
Find the maximum value of Vinitial so that the rooler coaster stays on the track at solely becasue of gravity. Answer in units of m/s.
Please help me with this problem, i have been tried to work it out many time but it wasn't correct

part b: Using the value of V(inital) calculated in question 1, determine the value of h' that is necessary if the roller coaster just makes it to point B. Answer in units of m.

Answer 1

You haven't given us enough information to solve the problem because we don't know how high point A is and we are not sure what you mean by "the roller coaster stays on the track solely because of gravity"

But, if you are dealing with a hill situation and the question is whether the roller coaster sticks to the track as it goes over the hill or leaves the track, here is what you have to consider.

At the top of the hill, roller coaster will have a certain velocity, call it Vh. The hill will have a certain radius of curvature, call it R.

If the roller coaster is going to stay on the track, it will have to experience an acceleration (the centripetal acceleration) downward (i.e. to the center of the circle defining the radius of curvature) which is at least equal to Vh/R.

But the roller experience an acceleration due to gravity, called g (9.8 m/s^2). Thus the question is, under what circumstance Vh/R = g. (i.e. if Vh/R is greater than g, the roller coaster will leave the track)

If there is no friction, total energy is conserved. In this case, this means that the sum of the potential and kinetic energies are constant:

P(t) + K(t) = constant.

K(t), the kinetic energy at time t, is (1/2) x M x V(t)^2

P(t), the potential energy at time t, is g x M x H(t)

We are given that H(0) is 26m. Suppose we also knew H(Th) where Th is the time when the roller coaster is at the top of the hill.

Then we would have:

P(0) + K(0) = P(Th) + K(Th) and
V(Th)/R = g

We know R, g. With the two heights we can compute P(0) and P(Th) (We ignore M because it cancels. Assume it is just 1)

Then we have two equations, in the two unknowns, V(Th) and V(0). V(0) is the answer you want.