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## Homework Statement

Based on the height of a first hill (114.5m), mathematically determine the minimum height (circular motion and radius) of the next hill based on the speed generated on the first hill.

At the bottom of the first hill, the velocity of the roller coaster is 47.38m/s. There is no friction force, so the velocity is the same at the bottom of the second hill.

There is a flat section of the track between the two hills that is 93.81m long. At the top of the second hill, when it is 112m tall, the velocity is 7.03m/s, and the roller coaster is able to go over and down the second hill safely.

## Homework Equations

KE = mgh

Fnet= N-W= -ma

a= v^2/r

## The Attempt at a Solution

When the hill isn't tall enough, the roller coaster launches off the top of the hill. When the hill is too tall, the coaster doesn't reach the top and slides back down.

I tried to calculate the maximum height,

KE (at bottom of the second hill) = mgh

0.5mv^2 = mgh

mass cancels

0.5v^2 = gh

H= (0.5v^2)/9.8

H= 114.5 m, which is the same height as the first hill so the coaster will be able to reach the top without sliding back down.

When I tried to get the minimum safe height, I had issues.

I tried summation of forces.

Y direction:

Fnet= -N-W = -ma (since acceleration is directed downward toward the middle of the hill)

The coaster flies off when the normal force = 0.

-W=-ma

Mg=ma

G=a

g= v^2/r

r= v^2/g

I plugged in the velocity at the bottom of the hill for v, and got 229.06 m, which exceeds the max safe height.

If it helps, it took 11.72s to reach the bottom of the second hill, then 1.98s to travel the flat section of the track, then 6.11s to reach the top of the second hill.

Any help is appreciated. Thanks.

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