Arrange the examples in order, starting with the object that has the least amount of energy at the top and the most amount of energy at the bottom. Assume no friction and use the following formulas:
[tex]\[ g = 9.8 \, m/s^2 \][/tex]
[tex]\[ PE = m \times g \times h \][/tex]
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]

- A book with a mass of 0.75 kilograms resting on a shelf at a height of 15 meters
- A brick with a mass of 25 kilograms falling with a velocity of 10 meters/second when it's 4 meters above ground
- A ball with a mass of 0.25 kilograms rolling



Answer :

To solve this problem, let's break it down into parts and calculate the different forms of energy for each object.

### 1. The Book
Parameters:
- Mass ([tex]\(m\)[/tex]) = 0.75 kg
- Height ([tex]\(h\)[/tex]) = 15 meters

Since the book is resting on a shelf, it has potential energy but no kinetic energy because it is not moving.

Potential Energy (PE):
[tex]\[ PE_{\text{book}} = m \times g \times h = 0.75 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 15 \, \text{m} \][/tex]

Calculations:
[tex]\[ PE_{\text{book}} = 110.25 \, \text{Joules} \][/tex]

### 2. The Brick
Parameters:
- Mass ([tex]\(m\)[/tex]) = 25 kg
- Height ([tex]\(h\)[/tex]) = 4 meters
- Velocity ([tex]\(v\)[/tex]) = 10 meters/second

The brick has both potential energy (because of its height) and kinetic energy (because it is falling).

Potential Energy (PE):
[tex]\[ PE_{\text{brick}} = m \times g \times h = 25 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 4 \, \text{m} \][/tex]

Calculations:
[tex]\[ PE_{\text{brick}} = 980 \, \text{Joules} \][/tex]

Kinetic Energy (KE):
[tex]\[ KE_{\text{brick}} = \frac{1}{2} \times m \times v^2 = \frac{1}{2} \times 25 \, \text{kg} \times (10 \, \text{m/s})^2 \][/tex]

Calculations:
[tex]\[ KE_{\text{brick}} = 1250 \, \text{Joules} \][/tex]

Total Energy:
[tex]\[ \text{Total Energy}_{\text{brick}} = PE_{\text{brick}} + KE_{\text{brick}} = 980 \, \text{Joules} + 1250 \, \text{Joules} \][/tex]

Calculations:
[tex]\[ \text{Total Energy}_{\text{brick}} = 2230 \, \text{Joules} \][/tex]

### 3. The Ball
Parameters:
- Mass ([tex]\(m\)[/tex]) = 0.25 kg
- Velocity ([tex]\(v\)[/tex]) = 0 meters/second (it's rolling on a flat surface)

The ball only has kinetic energy, as it is not elevated.

Kinetic Energy (KE):
[tex]\[ KE_{\text{ball}} = \frac{1}{2} \times m \times v^2 = \frac{1}{2} \times 0.25 \, \text{kg} \times (0 \, \text{m/s})^2 \][/tex]

Calculations:
[tex]\[ KE_{\text{ball}} = 0 \, \text{Joules} \][/tex]

### Arranging the Energies
Now that we have the energies calculated:

- Ball: [tex]\(0 \, \text{Joules}\)[/tex]
- Book: [tex]\(110.25 \, \text{Joules}\)[/tex]
- Brick: [tex]\(2230 \, \text{Joules}\)[/tex]

We can arrange the objects in order of increasing total energy:

1. A ball with a mass of 0.25 kilograms rolling ([tex]\(0 \, \text{Joules}\)[/tex])
2. A book with a mass of 0.75 kilograms resting on a shelf at a height of 15 meters ([tex]\(110.25 \, \text{Joules}\)[/tex])
3. A brick with a mass of 25 kilograms falling with a velocity of 10 meters/second when it's 4 meters above ground ([tex]\(2230 \, \text{Joules}\)[/tex])

Thus, the correct order, from least energy to most energy, is:
- Ball
- Book
- Brick