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Sagot :
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
### 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
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