Aroller coaster starts at the top of a hill of height h, goes down the hill, and does a circular loop of radius r before continuing on. this roller coaster is not attached to the track — it only glides on top of it, so it could fall off at the top of the loop. this section of the roller coaster is shown below.
if the roller coaster starts at the top of the hill with zero velocity, what is the expression for the velocity of the roller coaster at the top of the loop?
a. draw the free-body diagram for the roller coaster at the top of the loop. (1 point)

b. write the expression for the net force in the y-direction at the top of the loop, in terms of w, fn, m, v, and r. (1 point)

c. write the equation that will tell you the velocity at which the coaster will fall when it reaches the top of the loop. (1 point)

d. if the coaster has a mass of 50 kg and the loop has a radius of 30 m, how fast must the coaster be going at the top of the loop to keep it from falling? (1 point)

e. based on the values in the previous part, how high must the roller coaster start out to prevent it from falling from the top of the loop? (2 points)

f. if friction along the track caused the coaster to lose 10% of its initial energy by the time it reached the top of the loop, how high would h need to be to keep the coaster from falling off? (2 points)