a frictionless roller coaster car of mass m = 719 kg tops the first hill with speed v0 = 15.2 m/s at height h = 48.9 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?

First hill=h
A=h
B=h/2
C=ground level

To calculate the work done by the gravitational force on the roller coaster car at different points, we need to use the concept of gravitational potential energy and the work-energy principle.

a) Work from point H to point A:
The work done by the gravitational force is equal to the change in gravitational potential energy. The formula to calculate gravitational potential energy is given by:

Potential Energy = mass * gravity * height

In this case, the height is the difference in elevation between the starting point (h) and point A (h). Therefore, the potential energy at point A is:

Potential Energy at A = mass * gravity * (h - h)

Since h-h = 0, the potential energy at point A is zero. Hence, the work done by the gravitational force from point H to point A is also zero.

b) Work from point H to point B:
Similar to the previous case, the potential energy at point B can be calculated as:

Potential Energy at B = mass * gravity * (h - h/2)

The work done by the gravitational force from point H to point B is equal to the change in potential energy, which is:

Work from H to B = Potential Energy at B - Potential Energy at H
= (mass * gravity * (h - h/2)) - (mass * gravity * h)
= mass * gravity * ((h - h/2) - h/2)
= mass * gravity * (h/2 - h/2)
= 0

Therefore, the work done by the gravitational force from point H to point B is also zero.

c) Work from point H to point C:
Similar to the previous cases, the potential energy at point C can be calculated as:

Potential Energy at C = mass * gravity * (h - ground level)

In this case, the ground level is taken as the reference point, so the potential energy at point C is zero. Hence, the work done by the gravitational force from point H to point C is also zero.

d) Potential energy at point B:
Since the gravitational potential energy at point C is taken as zero, the potential energy at point B can be calculated as the difference between the potential energy at C and the potential energy at B:

Potential Energy at B = Potential Energy at C - Potential Energy at B
= 0 - (mass * gravity * (h - h/2))
= -mass * gravity * (h/2)

Therefore, the potential energy at point B is negative and equal to -mass * gravity * (h/2).

e) Potential energy at point A:
Using the same approach as above, the potential energy at point A can be calculated as the difference between the potential energy at C and the potential energy at A:

Potential Energy at A = Potential Energy at C - Potential Energy at A
= 0 - (mass * gravity * (h - h))
= -mass * gravity * 0
= 0

Therefore, the potential energy at point A is also zero.