Finding height of frictionless roller coaster

[Agent M27]

Homework Statement

Consider a frictionless roller coaster. The acceleration due to gravity is 9.8 ms^-2. Passenger cars start at point A with zero initial speed, accelerate as they go down to a point B, swing around the vertical circular loop B-C-B of radius 15m, then go on towards further adventures. When a car goes through the top of the loop, point C, the passengers feel weightless. What is the height of the starting point A above the loops bottom, point B?

Homework Equations

U_gi+KE_i=U_gf+KE_f

mgh_i+.5mv_i²=mgh_f+.5mv²_f

v$$\geq$$$$\sqrt{gr}$$

The Attempt at a Solution

Since the cart is at the top of the track and has zero initial speed the initial K.E. is canceled out. Also since the final location of the cart is the refrence point, the U_gf will be canceled out, so I am left with the following relationship:

mgh=.5mv²

V= 12.1244 m/s

h=$$\frac{v²}{2g}$$
=7.5m

This answer is clearly incorrect because the radius of the loop alone is 15m. Any clues as to where I am missing the mark? Thanks in advance.

Joe

 

[collinsmark]

Hello Joe,

=7.5m

This answer is clearly incorrect because the radius of the loop alone is 15m.

Not necessarily incorrect. How did you define your variable 'h'? Take a closer look at how you set up your equations (which might be just fine the way they are, btw. [Although you left out the centripetal force part, but is how I assume you arrived at "v = 12.1244 m/s".]). Did you define 'h' with respect to the ground, or with respect to some other point?

 

[Agent M27]

I thought I defined h with respect to the ground, which is why I assumed the final gravitational potential to equal 0. I also made the assumption my starting point had to be higher than the loop just based on intuition, but apparently it doesn't have to be? By the way you worded your reply I am thinking that maybe I found the distance between point C (the top of the loop) and the starting point, is that correct? Thanks again.

Joe

 

[Agent M27]

Solved it, 37.5m! The issue was that I some how defined my reference point as point C... How did I make that mistake, which assumption was incorrect? Thanks again collinsmark.

Joe

 

[collinsmark]

Hello Joe,

Solved it, 37.5m! The issue was that I some how defined my reference point as point C... How did I make that mistake, which assumption was incorrect? Thanks again collinsmark.

The only incorrect assumption was your:

mgh=.5mv²

That defines 'h' as the distance from point A to point C (point C is where v is being measured).

If you wanted to define 'h' as the height with respect to point B, the bottom of the loop, the equation should be,

(Potential energy at point A) = (kinetic energy at point C) + (potential energy at point C)
which is,

mgh = 0.5mv² + mg(30 m)