Question

In the figure, a frictionless roller coaster car of mass m = 637 kg tops the first hill with speed v₀ = 19.3 m/s at height h = 37.2 m. How much work does the gravitational force do on the car from that point to (a) point A, (b) point B, and (c) point C? If the gravitational potential energy of the car-Earth system is taken to be zero at C, what is its value when the car is at (d) B and (e) A?

In the figure, a frictionless roller coaster car of mass m = 637 kg tops the first hill with speed v0 = 19.3 m/s at height h = 37.2 m. How mu ch work does the gravitational force do on the car from that point to (a) point A, (b) point B, and © point C? If the gravitational potential energy of the car-Earth system is taken to be zero at C, what is its value when the car is at (d) B and (e) A?

Answer 1

Concepts and reason

The main concepts required to solve this problem are the work done, potential energy, gravitational force, mass and acceleration.

Initially, write the equation for the work done by the gravitational force, potential energy. Use these formulas and calculate the work done and potential energy for the given cases in the question.

Fundamentals

The equation for the gravitational force is,

Here, m is the mass of the object and g is the acceleration due to gravity.

The work done on an object due to the gravitational force is,

Here, h is the height of the object from the earth.

The equation for the potential energy on an object due to its position is,

Here, m is the mass of the object, g is the acceleration due to gravity and h is the height of the object.

(a)

The gravitational force acting on the car is,

The equation for the work done on the car at point A is,

But the vertical displacement of the car from its initial point to the point A is zero.

Substitute mg for , and 0 for s in above equation.

(b)

The displacement of the car from its initial position to the position at B is

The equation for the work done by the gravitational force on car at point B is,

Substitute 637 kg for m, for g, and 37.2 m for h in above equation.

(c)

The total vertical displacement of the car from its initial position to its final positional at C is h.

Therefore, the equation for the work done by the gravitational force on the car at point C is,

Substitute 637 kg for m, for g, and 37.2 m for h in above equation.

(d)

The height of the car at point B is h/2.

The equation for the gravitational potential energy at point B is,

Substitute 637 kg for m, for g, and 37.2 m for h in above equation.

(e)

The equation for the gravitational potential energy of the car at point A is,

Substitute 637 kg for m, for g, and 37.2 m for h in above equation.

Ans: Part a

The work done by the gravitational force on the car at point A is 0 J.

Part b

The work done by the gravitational force on the car at point B is 116 kJ.

Part c

The work done by the gravitational force on the car at point C is 232.2 kJ.

Part d

The gravitational potential energy of the car at point B is 116 kJ.

Part e

The gravitational potential energy of the car at point A is 232.2 kJ.