You wish to make a simple amusement park ride in which a steel-wheeled roller-coaster car travels down one long slope, where rolling friction is negligible, and later slows to a stop through kinetic friction between the roller coaster's locked wheels sliding along a horizontal plastic (polystyrene) track. Assume the roller-coaster car (filled with passengers) has a mass of 756.5 kg and starts 88.2 m above the ground. (a) Calculate how fast the car is going when it reaches the bottom of the hill.

The speed  of roller coaster at the ground is 41.6 m/s.

Explanation:

mass of roller coaster, m = 756.5 kg

height, h = 88.2 m

(a) Let the speed of car at the ground is v.

Use conservation of energy

Potential energy at height = kinetic energy at bottom

we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .