Answer:
m = maximum mass of the coaster = 410 kg
d = maximum spring compression = 2.3 m
h = maximum height of the track = 11 m
H = maximum difference in height of the track = 19 m
g = acceleration by gravity = 9.8 m/s²
k = spring constant (without safety margin) = ?
K = spring constant (with safety margin) = ?
V = maximum speed of the coaster = ?
The gravitational potential energy of the coaster on the top of the 11 m high hill (relative to its initial starting point) is:
PEg = m g h
PEg = (410 kg) (9.8 m/s²) (11 m)
PEg = 44198 J
To reach that height, the elastic potential energy stored in the spring must be the same, so:
PEg = PEe = k d² / 2
(44198 J) = k (2.3 m)² / 2
k = 16710 N/m
Adding 14% to that value, you get:
K = 1.14 (16710 N/m)
K = 19045 N/m - answer spring constant
When fully compressed, the elastic potential energy stored in the spring is:
PEe = K d² / 2
PEe = (19045 N/m) (2.3m)² / 2
PEe = 51326 J
The difference in height between the starting point and the lowest point of the track is:
Δh = H - h
Δh = (19 m) - (11 m)
Δh = 8 m
So the initial gravitational potential energy of 330 kg coaster, relative to the lowest point, is
PEg = m g Δh
PEg = (340 kg) (9.8 m/s) (8 m)
PEg = 26656 J
The total energy of the coaster at its starting point (again, relative to the lowest point) is:
TE = PEe + PEg
TE = (51326J) + (26656 J)
TE = 77982J
At the lowest point of the track, all that energy is converted to kinetic energy, so the speed at that point will be:
TE = KE = m V² / 2
(77982 J) = (340kg) V² / 2
V = 21.46 m/s - answer maximum speed