Max Velocity and Height of Roller Coaster Track

[Panphobia]

Homework Statement

A man starts riding on a frictionless roller coaster track at initial velocity v$_{o}$. At point A the radius of the bend is r.

a) What is the maximum v$_{o}$ such that when the cart gets at point A it stays on the track?
b) Using you answer in a) what should h' be such that the cart only just makes it to point B?

Homework Equations

Ei = Ef
mv^2/r

The Attempt at a Solution

So this question was kind of bugging me, and I want to know if I got it right so for the a) part this is what I did firstly

mgh + (1/2)mv$_{o}$$^{2}$ = mg(2/3)h+1/2mv$^{2}$
v = $\sqrt{2gh/3 + Vo^2}$

and then since at point be its a circle and its looking for the max velocity until the normal force = 0 I did
mv^2/r = mg
v = $\sqrt{rg}$
then
rg = 2gh/3 + v$_{o}$$^{2}$
v$_{o}$ = $\sqrt{g*(r-(2/3)h)}$THEN for b) I just used Energy again.
so
v = $\sqrt{rg}$
Ei = (1/2)mrg + mg(2/3)h
Ef = mgh'
h' = r/2 + 2h/3

 

[TSny]

Looks good to me. :thumbs:

 

[Panphobia]

Oh thank you so much!This was one of two questions I was unsure of so I just went with my gut on it.