You have been hired to design a spring-launched roller coaster that will carry t

Question

You have been hired to design a spring-launched roller coaster that will carry two passengers per car. The car goes up a 13-m-high hill, then descends 20m to the track's lowest point. You've determined that the spring can be compressed a maximum of 2.4m and that a loaded car will have a maximum mass of 450kg . For safety reasons, the spring constant should be 11% larger than the minimum needed for the car to just make it over the top.

What is the maximum speed of a 340kg car if the spring is compressed the full amount?

Explanation / Answer

1)
Set 0 potential at the bottom of the 13 m hill

PE at the top of the hill:
PE(grav.) = mgh = (450 kg)(9.8 m/s^2)(13 m) = 57330 J

PE of fully compressed spring:
PE(spring) = (1/2)kx^2
57330 J = (1/2)k'(2.4 m)^2
k' = 19906.25 N/m
k = 1.11(19906.25 N/m) = 22095.9 N/m

2)
Set 0 potential at the bottom of the 20 m slope

Initially:
ME = PE(grav) + PE(spring)
= 57330 J + 1.11(57330 J) = 120966.3 J

At the bottom of the 20-m slope:
ME = KE
KE = (1/2)mv^2
120966.3 J = (1/2)(340 kg)v^2
v = 26.67 m/s

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