A roller coaster\'s car rolls downhill from a high level and hten follows a vert

Question

A roller coaster's car rolls downhill from a high level and hten follows a vertical loup in the track. The car, with passangers, has a mass of 1680kg. The loop has its base on the ground and has as height of 14m. Assume there is no friction between the car and the track,

(a) What minimum initial height does the car need to have so that it can speed around the loop without falling to the ground? (The centripetal force at the top of the loop must be greater than the gravtitational force).

(b) If the car starts at the height you found in part (a), what will be its kinetic energy as it finishes the loop and returns to ground level?

Explanation / Answer

Let the minimum Initial height car need to have = h
Initial Potenital Energy = m*g*h


At top of the loop to maintain it's track
Fc = mv^2/r = m*g
v = sqrt(g*r)
where r = h/2 = 14/2 = 7m

v = sqrt(9.8*7) m/s
v = 8.3 m/s

Kinetic Energy at the top of the loop = 0.5*m*8.3^2

Potential Energy at the top of the loop = m*g*14

Total Energy at the bottom of the loop = 0.5*m*8.3^2 + m*g*14

Initial Potenital Energy = Total Energy at the bottom of the loop
m*g*h = 0.5*m*8.3^2 + m*g*14
9.8 * h = 0.5*8.3^2 + 9.8*14
h = 17.5 m

Minimum Initlal Height car need to have, h = 17.5 m

b)
As there is no friction present, there would be no loss of energy.
Therefore,
Kinetic Energy at the ground level = m*g*h
Kinetic Energy at the ground level = 1680 * 9.8 * 17.5 J
Kinetic Energy at the ground level = 2.88 * 10^5 J

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