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Sagot :
To calculate the kinetic energy (KE) of a bottle using the formula [tex]\( \text{KE} = \frac{1}{2}mv^2 \)[/tex], we will substitute the given values for the mass ([tex]\(m\)[/tex]) and the speed ([tex]\(v\)[/tex]). The speed is given as [tex]\(4 \text{ m/s}\)[/tex]. Let's compute the kinetic energy for each mass provided.
1. When the mass of the bottle is [tex]\(0.125 \text{ kg}\)[/tex]:
[tex]\[ \text{KE} = \frac{1}{2} \times 0.125 \text{ kg} \times (4 \text{ m/s})^2 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 0.125 \times 16 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 2 \][/tex]
[tex]\[ \text{KE} = 1.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2 \][/tex]
2. When the mass of the bottle is [tex]\(0.250 \text{ kg}\)[/tex]:
[tex]\[ \text{KE} = \frac{1}{2} \times 0.250 \text{ kg} \times (4 \text{ m/s})^2 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 0.250 \times 16 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 4 \][/tex]
[tex]\[ \text{KE} = 2.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2 \][/tex]
3. When the mass of the bottle is [tex]\(0.375 \text{ kg}\)[/tex]:
[tex]\[ \text{KE} = \frac{1}{2} \times 0.375 \text{ kg} \times (4 \text{ m/s})^2 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 0.375 \times 16 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 6 \][/tex]
[tex]\[ \text{KE} = 3.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2 \][/tex]
4. When the mass of the bottle is [tex]\(0.500 \text{ kg}\)[/tex]:
[tex]\[ \text{KE} = \frac{1}{2} \times 0.500 \text{ kg} \times (4 \text{ m/s})^2 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 0.500 \times 16 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 8 \][/tex]
[tex]\[ \text{KE} = 4.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2 \][/tex]
So, recording the calculations:
- When the mass of the bottle is [tex]\(0.125 \text{ kg}\)[/tex], the KE is [tex]\(1.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2\)[/tex].
- When the mass of the bottle is [tex]\(0.250 \text{ kg}\)[/tex], the KE is [tex]\(2.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2\)[/tex].
- When the mass of the bottle is [tex]\(0.375 \text{ kg}\)[/tex], the KE is [tex]\(3.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2\)[/tex].
- When the mass of the bottle is [tex]\(0.500 \text{ kg}\)[/tex], the KE is [tex]\(4.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2\)[/tex].
1. When the mass of the bottle is [tex]\(0.125 \text{ kg}\)[/tex]:
[tex]\[ \text{KE} = \frac{1}{2} \times 0.125 \text{ kg} \times (4 \text{ m/s})^2 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 0.125 \times 16 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 2 \][/tex]
[tex]\[ \text{KE} = 1.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2 \][/tex]
2. When the mass of the bottle is [tex]\(0.250 \text{ kg}\)[/tex]:
[tex]\[ \text{KE} = \frac{1}{2} \times 0.250 \text{ kg} \times (4 \text{ m/s})^2 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 0.250 \times 16 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 4 \][/tex]
[tex]\[ \text{KE} = 2.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2 \][/tex]
3. When the mass of the bottle is [tex]\(0.375 \text{ kg}\)[/tex]:
[tex]\[ \text{KE} = \frac{1}{2} \times 0.375 \text{ kg} \times (4 \text{ m/s})^2 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 0.375 \times 16 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 6 \][/tex]
[tex]\[ \text{KE} = 3.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2 \][/tex]
4. When the mass of the bottle is [tex]\(0.500 \text{ kg}\)[/tex]:
[tex]\[ \text{KE} = \frac{1}{2} \times 0.500 \text{ kg} \times (4 \text{ m/s})^2 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 0.500 \times 16 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 8 \][/tex]
[tex]\[ \text{KE} = 4.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2 \][/tex]
So, recording the calculations:
- When the mass of the bottle is [tex]\(0.125 \text{ kg}\)[/tex], the KE is [tex]\(1.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2\)[/tex].
- When the mass of the bottle is [tex]\(0.250 \text{ kg}\)[/tex], the KE is [tex]\(2.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2\)[/tex].
- When the mass of the bottle is [tex]\(0.375 \text{ kg}\)[/tex], the KE is [tex]\(3.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2\)[/tex].
- When the mass of the bottle is [tex]\(0.500 \text{ kg}\)[/tex], the KE is [tex]\(4.0 \text{ kg} \cdot \text{m}^2 / \text{s}^2\)[/tex].
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