A roller coaster powered by a spring launcher follows a frictionless track as shown below. The roller coaster cart has a mass of 500 kg, and the spring launcher has a spring constant of 1500 N/m.

Question

a. If the cart starts at a height h1 = 10m, how much must the spring be compressed for the cart to make it over the hill at h2 if h2=25m?
b. What is the speed of the cart when it reaches h3 if h3=5m (assume that the cart is launched as derived in part a)?
c. What is the total work done by gravity on the cart between h1 and h3?
d. What is the fastest speed the cart reaches anywhere on the track?

Answer 1

If H2 = 25
then the car must have potential energy of
m g H2 = 500(9.81)(25) = 122,625 Joules at start
so
122,625 = m g H1 + (1/2) k x^2
122,625 = 500*9.81*10 + (1/2) k x^2
(1/2) 1500 * x^2 = 73,575
so
x = 9.9 meters
part b later maybe

Answer 2

well at H2 it had total energy of 122,625.
now it goes down 20 meters to 5 meters off ground
so it loses m*9.81*20 joules of potential
so
(1/2) m v^2 = m * 9.81 *20
v= sqrt (2 * 9.81*20)
by the way if something falls from height h it is handy to remember that its speed reaches sqrt(2 g h)
v = 19.8 m/s

Answer 3

H1 = 10
H3 = 5
falls 5 meters in all
m g h = 500 * 9.81 * 5 = 24,525 Joules

Answer 4

The highest speed is at the lowest point.
I have to assume that is H3 of 5 meters
so it started with 122,625 Joules
it then dropped from the 10 meter start to the 5 meter point 3, gaining 24,525 Joules
so
(1/2) m v^2 = 122,625 + 24,525

v^2 = 2(147150) / 500
v = 24.3 m/s