An amusement park ride consists of a cart with riders (of total mass 200 kg) tha

Question

An amusement park ride consists of a cart with riders (of total mass 200 kg) that is set in motion by a large spring with spring constant 800 N/m. The cart travels along the flat horizontal section of track that is located 17 m above the ground and then down a ramp toward the loop-the-loop which starts at the ground and has an unknown diameter. The entire track is frictionless. Show your work on this one to get full credit! If the spring is initially compressed by 4 m, what is the speed of the cart as it leaves the spring? Is this speed reasonable? What is the speed of the cart at the bottom of the ramp before reaching the loop-the-loop? Is this speed reasonable? What is the maximum height of the loop-the-loop that the cart can do without falling? Is that a reasonable height?

Explanation / Answer

a) kx^2 /2 = mv^2 /2
800 x 4^2 = 200v^2
v = 8 m/s

b) using energy conservation,
mgh + mv^2 /2 = mv1^2 /2
9.8x17 + 8^2 /2   = v1^2 /2
v1 = 19.93 m/s

c) at max. height,
N + mg = mv^2 /2
just N = 0 case for minimum v
g = v^2 /2
v = 4.43m/s
so mv1^2 = mgh + m4.43^2 /2
19.93^2/2 = 9.8h + 4.43^2/2
h = 19.27 m

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