Question

Which roller coaster will NOT have enough kinetic energy at the bottom of the first hill to carry the car over the second hill?

Answer 1

I don’t have the picture or numbers from your problem, so I can’t say which named coaster fails. If you upload the diagram or give the heights/speeds I’ll pick the correct coaster.

Meanwhile, here’s the quick test you should apply (no friction or other losses assumed):

- Label the top of the first hill h1 and the top of the second hill h2 (heights measured from the same reference, e.g. the valley between them or ground).
- If the car starts from rest at the top of the first hill, conservation of energy gives its kinetic energy at the bottom as KE_bottom = m g h1 (if bottom height is zero) or equivalently the speed at the bottom v = sqrt(2 g (h1 − h_bottom)).
- To reach the top of the second hill the car needs potential energy m g h2 (plus any kinetic energy if it must still have speed there).
- So the necessary condition (frictionless) to just make it over is h1 ≥ h2. If h1 < h2 it cannot reach the second hill.

If friction or energy losses W_loss are present, use:
m g h1 − W_loss ≥ m g h2
or
h1 − h2 ≥ W_loss/(m g).

If you paste the heights or the problem figure, I’ll apply this and tell you which coaster will not make it.