A very large roller coaster has an initial hill that has an 85 m drop. Note that a roller coaster has
no engine. After being pulled up the first hill by a chain the train moves along the track with only
gravity, normal forces due to the track, friction and air drag acting on it. Let us approximate the
speed of the train at the top of the initial hill as zero (it is moving very slowly as it is released from
the chain).
(a) What is the maximum height of a loop placed after this drop (the maximum height that the
train can possibly return to after the drop)? Ignore friction and drag.
(b) If the loop were built to the height found in part a) what speed would the roller coaster train
be going as it passed over the top of the loop? Draw an FBD for a car of the roller coaster
for the instant it passes over the top of the loop. What would a rider in the roller coaster feel
as they go over the top?
(c) If the loop is very small compared to the hill the roller coaster will be going very fast as it
passes over the top of the loop. Draw an FBD for a car of the roller coaster for the instant it
passes over the top of the loop. What would a rider in the roller coaster feel as they go over
the top in this case?
(d) What is the maximum height of the loop if the riders are not to feel that they are “hanging”
from their restraints as they pass over the top of the loop?
(e) Up to this point in the problem we have assumed that there is no friction. Now suppose
that there is a coefficient of rolling friction between the wheels and track of 0.050. Suppose
the initial hill is a uniform slope with a 45◦ angle. Find the speed at the bottom of the hill,
accounting for friction. Compare this to what the speed would have been without friction.
(f) Bonus: Now assume that the first loop is the height you found in part (d) and that the
loop starts right at the bottom of the first hill. Find the height at the top of the first loop
accounting for friction on the hill and in the loop. Accounting for friction in the loop is very
hard, and you will only be able to do it approximately. To make it a bit easier you can assume
that the loop is circular (which is unrealistic - real roller coaster loops are elliptical).