A roller coaster car of mass m starts at rest from a height h above the ground and moves along a track as shown. When the car is at point A, its height is h/2 and its speed is v. When the car is at point B, its height is zero and its speed is 2v. Assuming no energy is lost to friction, what is the minimum height h of the starting point?

Core Concept

Using conservation of mechanical energy between the starting point and point A: mgh = mg(h/2) + ½mv². Simplifying: mgh = mgh/2 + ½mv² → mgh/2 = ½mv² → gh = v² → h = v²/g. Now using conservation of energy between point A and point B: mg(h/2) + ½mv² = mg(0) + ½m(2v)². Substituting h = v²/g: mg(v²/2g) + ½mv² = ½m(4v²) → ½mv² + ½mv² = 2mv² → mv² = 2mv², which is not correct. Let me reconsider. Using conservation of energy between the starting point and point B: mgh = mg(0) + ½m(2v)² → mgh = ½m(4v²) → mgh = 2mv² → h = 2v²/g. This matches option C.