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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.

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].

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].

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].

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 solve this problem, we will use the given formula for kinetic energy:

[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]

where:
- [tex]\( KE \)[/tex] is the kinetic energy,
- [tex]\( m \)[/tex] is the mass of the bottle,
- [tex]\( v \)[/tex] is the speed of the bottle.

We are given that the speed [tex]\( v \)[/tex] of the soda bottle is [tex]\( 4 \, \text{m/s} \)[/tex], and we need to calculate the kinetic energy for various masses.

### Step-by-Step Calculations

1. When the mass of the bottle is [tex]\( 0.125 \, \text{kg} \)[/tex]:
[tex]\[ KE = \frac{1}{2} \times 0.125 \, \text{kg} \times (4 \, \text{m/s})^2 \][/tex]
[tex]\[ KE = \frac{1}{2} \times 0.125 \, \times 16 \][/tex]
[tex]\[ KE = 1.0 \, \text{kg} \cdot \text{m}^2 / \text{s}^2 \][/tex]

2. When the mass of the bottle is [tex]\( 0.250 \, \text{kg} \)[/tex]:
[tex]\[ KE = \frac{1}{2} \times 0.250 \, \text{kg} \times (4 \, \text{m/s})^2 \][/tex]
[tex]\[ KE = \frac{1}{2} \times 0.250 \, \times 16 \][/tex]
[tex]\[ KE = 2.0 \, \text{kg} \cdot \text{m}^2 / \text{s}^2 \][/tex]

3. When the mass of the bottle is [tex]\( 0.375 \, \text{kg} \)[/tex]:
[tex]\[ KE = \frac{1}{2} \times 0.375 \, \text{kg} \times (4 \, \text{m/s})^2 \][/tex]
[tex]\[ KE = \frac{1}{2} \times 0.375 \, \times 16 \][/tex]
[tex]\[ KE = 3.0 \, \text{kg} \cdot \text{m}^2 / \text{s}^2 \][/tex]

4. When the mass of the bottle is [tex]\( 0.500 \, \text{kg} \)[/tex]:
[tex]\[ KE = \frac{1}{2} \times 0.500 \, \text{kg} \times (4 \, \text{m/s})^2 \][/tex]
[tex]\[ KE = \frac{1}{2} \times 0.500 \, \times 16 \][/tex]
[tex]\[ KE = 4.0 \, \text{kg} \cdot \text{m}^2 / \text{s}^2 \][/tex]

### Results Summary

- When the mass of the bottle is [tex]\( 0.125 \, \text{kg} \)[/tex], the [tex]\( KE \)[/tex] 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 [tex]\( KE \)[/tex] 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 [tex]\( KE \)[/tex] 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 [tex]\( KE \)[/tex] is [tex]\( 4.0 \, \text{kg} \cdot \text{m}^2 / \text{s}^2 \)[/tex].

By these calculations, we can observe how the kinetic energy of the bottle changes as its mass increases.
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