# What is the highest height a hill could be if the roller coaster is to go over the top of it?

## I am studying for my physics final and need help with one of the questions from our midterm: A roller coaster, which has a mass of 5,000 kg (people riding + train), is on a near frictionless track. It is 10.54 m above the ground and has a velocity (at this point) of 4.26 m/s. You are coming up on a hill that will go up and then brings you all the way down to ground level. What is the highest height the hill you are coming up on could be, if the coaster is to go over the top? I tried using the relationship of KE=PE, because I know energy is conserved, but that didn't work. What do I do?

Jul 30, 2018

That has to work if energy is conserved.

$\frac{1}{2} m {v}^{2} = m g \Delta h$, where $\Delta h$ is the height relative to train's current height

$\therefore \Delta h = \frac{1}{2} {v}^{2} / g$

The actual height $H$ is therefore:

• $H = 10.54 + \frac{1}{2} \cdot {4.26}^{2} / 9.81 \approx 11.5 \text{ m}$

And $\Delta h \approx 0.92 \text{ m}$