Roller Coaster

Solution:

1. This is a simple application of the conservation of energy law;

EPi + EKi = EPf + EKf

Note that in the absence of friction, the mass term can cancel out of the equation as it is in every term,

hf  is considered zero elevation, or the reference point point and thus has the value of  zero. The value of hi is the sum of 65 m and 20 m, so the equation for vf is given by

2. A free body diagram of the car at the bottom of the hill, where the radius of curvature for the track is 60 m, and the velocity is as calculated above yields the following equation.

3. This is once again a conservation of energy problem where hi is considered zero elevation, or the reference point point, thus having the value of zero. The value of hf  is 60 m, so the equation for vf is given by

4. When the car is in the banked turn it is the x-component of the normal force, FNx, that supplies the unbalanced force towards the center of the circle, and is therefore the centripetal force. This means that the y-component of the normal force, FNy, and weight, W, must be balanced. Division of the resulting equations allows one of the unknowns, namely FN, to be cancelled, leaving the variable theta.

5. This is once again a conservation of energy problem where h is considered zero elevation at the bottom of the loop, thus having the value of zero, however at the top of the loop hf  is 30 m. The value of hi is the sum of 60 m and 10 m

The equation for vf  at the bottom of the loop is given by,

The equation for vf  at the top of the loop is given by,

6. A free body diagram of the car at the top of the loop, where the radius of the loop is 15 m, and the velocity is as calculated above yields the following equation.