Spring-launched rollercoaster problem

[mst3kjunkie]

49. You have been hired to design a spring launched roller coaster that will carry two passengers per car. Te car goes up a 10m high hill, then descends 15m 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 400 kg. For safety reasons, the spring constant should be 10% 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 350 kg car if the spring is compressed the full amount?

I'm having trouble with part a. Here's what I've got so far:

known:
m=400kg

y0= 5m
y1= 15m
y2= 0m

v1= 0 m/s

s0= -2.0m (compression of spring)
s1= 0m

to find:
k
v0

.5mv0^2+mgy0+.5kS0^2 = .5mv1^2+mgy1+.5kS1^2

200v0^2+19600-2k = 58800

am I on the right track? how do I find out what I need to get to solve for part a of the problem?

 

[OlderDan]

Your equation is OK if it is interpreted correctly. On the left hand side the velocity is zero when the spring is compressed 2 meters. (There is no minus sign in front of the 2k in the next line.) On the right hand side, the spring is not compressed and if the minimum (no safety factor) k is used, the velocity is zero. Only the gravitational potential energy term survives. Solve for k and then add the 10% safety factor.

 

[kyle8921]

Yes, you are on the right track. To find the spring constant (k), you can use the equation you have written and solve for k. You can also use Hooke's Law, which states that the force exerted by a spring is directly proportional to the amount it is stretched or compressed. This can be written as F = -kx, where F is the force, k is the spring constant, and x is the displacement. In this case, the maximum displacement is 2.0m, so you can plug in the values for mass (400kg), acceleration due to gravity (9.8 m/s^2), and displacement (2.0m) to solve for k. Once you have the value for k, you can then use the equation you have written to solve for the maximum speed of the car (v0) when the spring is compressed by the full amount.