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Calculate the change in the kinetic energy (KE) of the bottle when the mass is increased. Use the formula [tex]KE = \frac{1}{2} mv^2[/tex], where [tex]m[/tex] is the mass and [tex]v[/tex] is the speed (velocity). Assume that the speed of the soda bottle falling from a height of [tex]0.8 \, m[/tex] will be [tex]4 \, m/s[/tex], and use this speed for each calculation.

Record your calculations in Table A of your Student Guide.

1. When the mass of the bottle is [tex]0.125 \, kg[/tex], the [tex]KE[/tex] is [tex]\square \, kg \cdot m^2 / s^2[/tex].
2. When the mass of the bottle is [tex]0.250 \, kg[/tex], the [tex]KE[/tex] is [tex]\square \, kg \cdot m^2 / s^2[/tex].
3. When the mass of the bottle is [tex]0.375 \, kg[/tex], the [tex]KE[/tex] is [tex]\square \, kg \cdot m^2 / s^2[/tex].
4. When the mass of the bottle is [tex]0.500 \, kg[/tex], the [tex]KE[/tex] is [tex]\square \, kg \cdot m^2 / s^2[/tex].

Sagot :

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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].
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