You are working with a team that is designing a new roller coaster-type amusement park ride for a major theme park. You are present for the testing of the ride, in which an empty 220 kg car is sent along the entire ride. Near the end of the ride, the car is at near rest at the top of a 100 m tall track. It then enters a final section, rolling down an undulating hill to ground level. The total length of track for this final section from the top to the ground is 250 m. For the first 230 m, a constant friction force of 400 N acts from computer-controlled brakes. For the last 20 m, which is horizontal at ground level, the computer increases the friction force to a value required for the speed to be reduced to zero just as the car arrives at the point on the track at which the passengers exit. (a)
Determine the required constant friction force (in N) for the last 20 m for the empty test car.
N
(b)
Find the highest speed (in m/s) reached by the car during the final section of track length 250 m.
m/s
(c)
You are asked by your team supervisor to determine the answers to parts (a) and (b) for a fully loaded car with an upper limit of 450 kg of passenger mass. Find these new values. (Give the friction force in N and the speed in m/s. Round the friction force to at least three significant figures.)
friction force for last 20 m
N
highest speed reached
m/s
(d)
The required friction force in part (c) is well within design limits. The fastest speed, however, is well below that of current leading rides, so you would like to increase the maximum speed. You can't make the tower taller above ground, so you decide to include a feature where part of the track goes underground. Determine the depth (in m) to which the underground part of the ride must go to increase the maximum speed to 53.0 m/s. Assume the overall length of the first part of the track remains at 230 m and the length of track from the top to the lowest underground point is 150 m. The same 400 N friction force acts on the entire 230 m section of track.
m
(e)
Is the construction in part (d) feasible?
Yes, there is enough track both for the depth requirement to be satisfied and for the horizontal displacement of the length of track from the top to the lowest underground point to be of the same order of magnitude as its height.
No, there is not enough track both for the depth requirement to be satisfied and for the horizontal displacement of the length of track from the top to the lowest underground point to be of the same order of magnitude as its height.