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: Recall energy - conservation principle

The total mechanical energy \(E = PE+KE\) (where \(PE\) is potential energy and \(KE\) is kinetic energy) is conserved in the absence of non - conservative forces (like friction, air resistance is neglected here). At the top of the hill, the roller - coaster car has maximum potential energy \(PE = mgh\) (where \(m\) is mass, \(g\) is gravitational acceleration, \(h\) is height) and zero kinetic energy (\(KE = 0\)).

Step2: Analyze halfway down the hill

Halfway down the hill, the height \(h\) is reduced to \(\frac{h}{2}\). So the potential energy becomes \(PE'=mg\frac{h}{2}=\frac{1}{2}mgh\). Since the total mechanical energy is conserved, the loss in potential energy \(\Delta PE=mgh - mg\frac{h}{2}=\frac{1}{2}mgh\) is converted into kinetic energy.

Answer:

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