Work Energy Theorem and Circular Motion

[Ilsem]

Homework Statement

A car is coasting without friction toward a hill of height 'h' and radius of curvature 'r'.
What initial speed will result in the car's wheels just losing contact with the roadway as the car crests the hill?

Homework Equations

Kinetic Energy = (1/2)(m)(v^2)
Potential Energy = mgy

The Attempt at a Solution

Because the acceleration will vary with time, constant energy kinematics can't be used to solve the question. Without friction, there are no nonconservative forces acting on the system, so there is no energy lost. Therefore the kinetic energy of the car at the bottom of the hill must be equal(?) to the potential energy of the car at the top of the hill. But I can't seem to work the radius of curvature into the theory at all. There's can't be a centripidal force because the only force holding the car to the curvature of the hill is the force of gravity. I'm a little stuck at this point. Thank you in advance for any help someone can give.

 

[LowlyPion]

Welcome to PF.

When the mv²/r is greater than mg contact force won't that mean that the car will lose contact with the road? So when v² = g*r.

So your initial mV² must be greater than m*g*h +mv²/2 where v² can be substituted with g*r ?

 

[Ilsem]

Thank you very much for the kind welcome.

I see. So if the formula a=(v^2)/r is usually used for centripidal force, and the only force holding the car to the curve is gravity, then this formula has to be used as a minimum where a=g? I think that makes sense.