A roller coaster car of mass m starts at rest from a height h above the ground and moves along a track with negligible friction. At the bottom of the first hill, it has a speed v. What is the maximum height it can reach on the next hill if it continues with the same speed?

Core Concept

By conservation of mechanical energy, the total energy at the starting point (mgh) equals the total energy at any other point. At the bottom of the first hill, the energy is (1/2)mv². For the car to reach the same height h on the next hill, it would need to have potential energy mgh, which requires all the kinetic energy to be converted back to potential energy. Since energy is conserved and there's no friction, the car can reach the same height h.