You have been hired to make a roller coaster more exciting. The owners want the speed at the bottom of the first hill doubled. How much higher must the first hill be built. Show all work and calculations

Question

jamesj · Answer

Well, suppose you drop an object, drive a go-cart, push a roller-coaster cart down a slope that has a vertical height of h.  At the bottom, all of the gravitational potential energy, PE, is converted into kinetic energy KE.  

Now PE at the top, at a height h is
$$ PE = mgh $$
At the bottom, the kinetic energy is
$$KE = \frac{1}{2}mv^2 $$
where v is speed.

Hence if all the PE from the top is converted into KE, we have
PE = KE
$$ mgh = \frac{1}{2}mv^2 $$
rearranging:
$$ v = \sqrt{2gh} $$

Now, given that, what do you have to modify h to be so that the resulting speed is 2v?  I.e., for what h' is
$$ \sqrt{2gh'} = 2v $$
?

That's what your question is asking.  What is h' in terms of h?

jamesj · Answer

Make sense?

anonymous · Answer

but How much higher must the first hill be built in order for the speed at the bottom of the first hill to be doubled?

jamesj · Answer

Right.  That's exactly what I'm setting up for you in the equations.  The new height h' is what you need to find, and express it in terms of the original height h.

So use those last two equations to solve for h' in terms of h.

jamesj · Answer

First, do those equations make sense?  In other words, do you understand the Physics of where they came from?  That's the most important thing.

What happens now to solve for h' in terms of h is algebra, pure mathematics.  That's of secondary importance, but I'll help you if you're stuck with it.

mani_jha · Answer

If you want to get a higher velocity, you must have a greater depth. So, should not you be asking how much lower it should be built? If it is higher than the previous one, you get a lower velocity, and you will probably be fired for that!

jamesj · Answer

The important thing is the difference in height of the first 'hill' vs. the first 'dip'; that's the variable h (and h') here.  

You can either raise the beginning ('hill') or lower the end ('dip'); it doesn't actually matter which for the purposes of raising the speed at the bottom of the dip.  But as the question sets it up: "How much higher must the first hill be built?"