Question

You have been hired to design a spring-launched roller coaster that will carry two passengers per car. The car goes up a 12-m-high hill, then descends 20m to the track's lowest point. You've determined that the spring can be compressed a maximum of 2.5m and that a loaded car will have a maximum mass of 450kg . For safety reasons, the spring constant should be 15% larger than the minimum needed for the car to just make it over the top.

A) What spring constant should you specify?

B) What is the maximum speed of a 360kg car if the spring is compressed the full amount?

Answer 1

Here is what I solved before, please modify the figures as per your question. Please let me know if you have further questions. Ifthis helps then kindly rate 5-stars.

You have been hired to design a spring-launched roller coaster that will carry two passengers per car. The car goes up a 12-m-high hill, then descends 18 m to the track's lowest point. You've determined that the spring can be compressed a maximum of 2.0 m and that a loaded car will have a maximum mass of 440 kg. For safety reasons, the spring constant should be 12 % larger than the minimum needed for the car to just make it over the top.
What spring constant should you specify?
k = ? N/m
What is the maximum speed of a 360 kg car if the spring is compressed the full amount?
v = ? m/s   

answer

conserving energy,

spring energy will be (1/2)k*x²

gravitational potential energy mgh

balancing,

(1/2)k*x²=mgh

considering max. compression of spring and max. weight

ie. x=2m, m=440 kg , h=12m

considering the condition where velocity just becomes 0 at the top of the cliff,

k=25872 N/m

the value of k₀ should be 1.12 times greater than k

therefore k₀=28976.64 N/m

considering the lowest point of the cliff..

conserving energy,

(1/2)k*x²=-m₁gh₂+(1/2)m₁v²

h₂=6

since the reference is at an axis which is 12 m from the top of the cliff.

m₁=360 kg

substituting the values,

v=20.965 m/sec