# Uniform Circular Motion roller coaster

This is another problem a buddy of mine and I cannot figure out if our life depended on it. Here's the question word for word:

A roller coaster track has a hill with a circular curve of radius 20 m. Find the speed of the roller coaster at the top of this hill.

Started by drawing a picture and writing down the given radius. Then we hit a wall.

Our first thought was how can we figured the speed of the roller coaster by only having the radius of the curve? We're looking through all our equations for this chapter and a couple integrate velocity and radius. Such as; v = 2(pie)r/T (T is the period); Ac = v^2/r; Fc = mv^2/r

Each equation have at least 2 unknowns. We hit another wall with these equations.

Please offer any kind of suggestions. Our ears and eyes are wide open.

TIA, Bryan

Tide
Homework Helper
That's actually a somewhat complicated problem. If it were a single car then simple energy conservation will give you the speed at the top of the loop. However, in a typical roller coaster, you have many cars and as the first one reaches the top there are still cars near the other end that are still on level rail. Energy is still conserved but you must include the mass of all the cars and their potential energy including all the cars that are part way up the loop.

It's somewhat more complicated if you want to know the speed at the top for cars that are not at either end.

The question only includes a single car on the track. And we can't use energy to solve this question because this question isn't from the energy chapter.

Doc Al
Mentor
As the problem is stated, there is not enough information to solve for the speed. Did you leave anything out? Any initial conditions? A typical kind of question to ask is "What is the maximum speed the car can have at the top of the hill without leaving the track?". That problem is solvable.

Nope, I didn't leave anything out. The question I wrote here is verbatim to the question on our piece of paper.

We even went as far as using equations for Satellites in Orbits because the only unknown in those equations were T, the period. But after doing all the math, the velocity of this cart was 1.05 x 10^5 m/s and I really don't think that's the case. lol

After searching Google for a couple minutes, I found a similiar problem. However, this problem deals with a loop di loop and at the top of the loop, the riders are upside down and feel weightless.

Using Centripetal Force;

Fc = Fg = w = mg
Fc = mv^2/r = mg
mv^2 = mgr
v^2 = gr
v = (sq.rt)(gr)
v = (sq.rt)(9.8m/s^2)(20m)
v = (sq.rt)(196 m^2/s^2)
v = 14 m/s

That seems like a more feasible answer. We don't have any solutions here to verify the correct answer. So we are just going on a whim. But would this make sense?

Thanks

Doc Al
Mentor
Again, the question as stated is incomplete. The solution you found is the answer to the "typical kind of question" that I posed in post #4.

Alright, then we're going with that answer. I mean, this is a college course, but it is by no means a Senior or Graduate level course. So I doubt our teacher would make anything too challenging and neglect to give us needed information. Thank you for your help though; it got me thinking more.