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

height of hill = h = 13 m

m = maximum mass of loaded car = 450 kg

so Potential energy at the Top ,

PE = mgh = (450 kg) ( 9.8) ( 13 m)

PE = 57330 J

X = compression of spring = 2.4 m

the spring potential energy is same as potential energy

so spring potential energy = Potential energy

spring potential energy is given as :: 1/2 kX^2

so 1/2 kX^2 = 57330                           ( since PE = 57330 )

(0.5) k (2.4)^2 = 57330

k = 19906.25

the spring constant should be 11% larger than the minimum needed

so Knew = K + 0.11 K = 1.11 K

Knew = 1.11 k

Knew = 1.11 (19906.25) = 22095.94 N/m

at the top , the total energy can be given as

Total energy = PE + Spring PE

Total energy = 57330 J + 1/2 (22095.94) (2.4)2 = 120966.31 J

at the bottom ::

total energy = kinetic energy

1/2 m V^2 = 120966.31 J

(0.5) (450) V^2 = 120966.31

V = 23.19 m/s

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