Question

half way down the hill, what could be said about the potential energy and kinetic energy of the roller coaster car? all of the potential energy is gone half of the original potential energy has been converted into kinetic energy

Explanation:

Step1: Analyze potential energy formula

The formula for gravitational potential energy is \(U = mgh\), where \(m\) is mass, \(g\) is the acceleration due to gravity, and \(h\) is height. At the top of the hill, the roller - coaster car has maximum potential energy \(U_{top}=mgh_{top}\). Half - way down the hill, \(h = \frac{h_{top}}{2}\), so the potential energy \(U_{half}=mgh_{half}=mg\frac{h_{top}}{2}=\frac{U_{top}}{2}\).

Step2: Consider energy conservation (ignoring non - conservative forces like friction for simplicity)

By the law of conservation of mechanical energy \(E = U + K\) (where \(E\) is the total mechanical energy, \(U\) is potential energy and \(K\) is kinetic energy). At the top of the hill \(E = U_{top}+K_{top}\), assuming \(K_{top} = 0\) (starts from rest), \(E = U_{top}\). Half - way down \(E=U_{half}+K_{half}\). Since \(E = U_{top}\) and \(U_{half}=\frac{U_{top}}{2}\), then \(K_{half}=\frac{U_{top}}{2}\)

Answer:

half of the original potential energy has been converted into kinetic energy