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Determine the required height h of the roller coaster so that when it is essentially at rest at the crest of the hill, it will reach a speed of 107 km/h when it comes to the bottom?

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Final answer:

The required height h of the roller coaster is approximately 114 m.

Step-by-step explanation:

To determine the required height h of the roller coaster, we can use the principle of conservation of energy. At the crest of the hill, the roller coaster is essentially at rest, so its initial kinetic energy is zero. At the bottom of the hill, the roller coaster has a speed of 107 km/h. Using the principle of conservation of energy, we can equate the initial gravitational potential energy to the final kinetic energy:

Initial gravitational potential energy = Final kinetic energy

mgh = 0.5mv^2

Where m is the mass of the roller coaster, g is the acceleration due to gravity, h is the height of the roller coaster, and v is the final speed of the roller coaster.

Given that the roller coaster starts from rest at the crest of the hill, its initial speed is zero. Converting the final speed from km/h to m/s:

v = (107 km/h) * (1000 m/1 km) * (1 h/3600 s) = 29.72 m/s

Plugging in the values, we can solve for h:

mgh = 0.5mv^2

(850 kg)(9.8 m/s^2)h = 0.5(850 kg)(29.72 m/s)^2

h = (0.5(850 kg)(29.72 m/s)^2)/(850 kg)(9.8 m/s^2)

h = 114.44 m

Therefore, the required height h of the roller coaster is approximately 114 m.

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