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To determine which marble has the highest kinetic energy at the bottom of the ramp, we can use the kinetic energy formula:
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
Here, [tex]\( KE \)[/tex] stands for kinetic energy, [tex]\( m \)[/tex] is the mass of the marble, and [tex]\( v \)[/tex] is the speed of the marble. Given that each marble reaches the bottom of the ramp at a speed of 3 meters per second ([tex]\( v = 3 \, \text{m/s} \)[/tex]), we need to evaluate the kinetic energy for each marble using their respective masses. The masses are provided in grams, so we will first convert them into kilograms by dividing by 1000. The table provides the following masses:
- Marble 1: 10 g
- Marble 2: 20 g
- Marble 3: 25 g
- Marble 4: 40 g
- Marble 5: 30 g
Let's calculate the masses in kilograms:
- Marble 1: [tex]\( 10 \, \text{g} = \frac{10}{1000} = 0.01 \, \text{kg} \)[/tex]
- Marble 2: [tex]\( 20 \, \text{g} = \frac{20}{1000} = 0.02 \, \text{kg} \)[/tex]
- Marble 3: [tex]\( 25 \, \text{g} = \frac{25}{1000} = 0.025 \, \text{kg} \)[/tex]
- Marble 4: [tex]\( 40 \, \text{g} = \frac{40}{1000} = 0.04 \, \text{kg} \)[/tex]
- Marble 5: [tex]\( 30 \, \text{g} = \frac{30}{1000} = 0.03 \, \text{kg} \)[/tex]
Next, we will calculate the kinetic energy for each marble:
1. Marble 1:
[tex]\[ KE_1 = \frac{1}{2} \times 0.01 \, \text{kg} \times (3 \, \text{m/s})^2 \][/tex]
[tex]\[ KE_1 = \frac{1}{2} \times 0.01 \times 9 \][/tex]
[tex]\[ KE_1 = 0.5 \times 0.01 \times 9 \][/tex]
[tex]\[ KE_1 = 0.045 \, \text{J} \][/tex]
2. Marble 2:
[tex]\[ KE_2 = \frac{1}{2} \times 0.02 \, \text{kg} \times (3 \, \text{m/s})^2 \][/tex]
[tex]\[ KE_2 = \frac{1}{2} \times 0.02 \times 9 \][/tex]
[tex]\[ KE_2 = 0.5 \times 0.02 \times 9 \][/tex]
[tex]\[ KE_2 = 0.09 \, \text{J} \][/tex]
3. Marble 3:
[tex]\[ KE_3 = \frac{1}{2} \times 0.025 \, \text{kg} \times (3 \, \text{m/s})^2 \][/tex]
[tex]\[ KE_3 = \frac{1}{2} \times 0.025 \times 9 \][/tex]
[tex]\[ KE_3 = 0.5 \times 0.025 \times 9 \][/tex]
[tex]\[ KE_3 = 0.1125 \, \text{J} \][/tex]
4. Marble 4:
[tex]\[ KE_4 = \frac{1}{2} \times 0.04 \, \text{kg} \times (3 \, \text{m/s})^2 \][/tex]
[tex]\[ KE_4 = \frac{1}{2} \times 0.04 \times 9 \][/tex]
[tex]\[ KE_4 = 0.5 \times 0.04 \times 9 \][/tex]
[tex]\[ KE_4 = 0.18 \, \text{J} \][/tex]
5. Marble 5:
[tex]\[ KE_5 = \frac{1}{2} \times 0.03 \, \text{kg} \times (3 \, \text{m/s})^2 \][/tex]
[tex]\[ KE_5 = \frac{1}{2} \times 0.03 \times 9 \][/tex]
[tex]\[ KE_5 = 0.5 \times 0.03 \times 9 \][/tex]
[tex]\[ KE_5 = 0.135 \, \text{J} \][/tex]
By comparing the kinetic energy values, we can see that Marble 4, with a kinetic energy of [tex]\( 0.18 \, \text{J} \)[/tex], has the highest kinetic energy.
Therefore, the correct answer is:
D. Marble 4
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
Here, [tex]\( KE \)[/tex] stands for kinetic energy, [tex]\( m \)[/tex] is the mass of the marble, and [tex]\( v \)[/tex] is the speed of the marble. Given that each marble reaches the bottom of the ramp at a speed of 3 meters per second ([tex]\( v = 3 \, \text{m/s} \)[/tex]), we need to evaluate the kinetic energy for each marble using their respective masses. The masses are provided in grams, so we will first convert them into kilograms by dividing by 1000. The table provides the following masses:
- Marble 1: 10 g
- Marble 2: 20 g
- Marble 3: 25 g
- Marble 4: 40 g
- Marble 5: 30 g
Let's calculate the masses in kilograms:
- Marble 1: [tex]\( 10 \, \text{g} = \frac{10}{1000} = 0.01 \, \text{kg} \)[/tex]
- Marble 2: [tex]\( 20 \, \text{g} = \frac{20}{1000} = 0.02 \, \text{kg} \)[/tex]
- Marble 3: [tex]\( 25 \, \text{g} = \frac{25}{1000} = 0.025 \, \text{kg} \)[/tex]
- Marble 4: [tex]\( 40 \, \text{g} = \frac{40}{1000} = 0.04 \, \text{kg} \)[/tex]
- Marble 5: [tex]\( 30 \, \text{g} = \frac{30}{1000} = 0.03 \, \text{kg} \)[/tex]
Next, we will calculate the kinetic energy for each marble:
1. Marble 1:
[tex]\[ KE_1 = \frac{1}{2} \times 0.01 \, \text{kg} \times (3 \, \text{m/s})^2 \][/tex]
[tex]\[ KE_1 = \frac{1}{2} \times 0.01 \times 9 \][/tex]
[tex]\[ KE_1 = 0.5 \times 0.01 \times 9 \][/tex]
[tex]\[ KE_1 = 0.045 \, \text{J} \][/tex]
2. Marble 2:
[tex]\[ KE_2 = \frac{1}{2} \times 0.02 \, \text{kg} \times (3 \, \text{m/s})^2 \][/tex]
[tex]\[ KE_2 = \frac{1}{2} \times 0.02 \times 9 \][/tex]
[tex]\[ KE_2 = 0.5 \times 0.02 \times 9 \][/tex]
[tex]\[ KE_2 = 0.09 \, \text{J} \][/tex]
3. Marble 3:
[tex]\[ KE_3 = \frac{1}{2} \times 0.025 \, \text{kg} \times (3 \, \text{m/s})^2 \][/tex]
[tex]\[ KE_3 = \frac{1}{2} \times 0.025 \times 9 \][/tex]
[tex]\[ KE_3 = 0.5 \times 0.025 \times 9 \][/tex]
[tex]\[ KE_3 = 0.1125 \, \text{J} \][/tex]
4. Marble 4:
[tex]\[ KE_4 = \frac{1}{2} \times 0.04 \, \text{kg} \times (3 \, \text{m/s})^2 \][/tex]
[tex]\[ KE_4 = \frac{1}{2} \times 0.04 \times 9 \][/tex]
[tex]\[ KE_4 = 0.5 \times 0.04 \times 9 \][/tex]
[tex]\[ KE_4 = 0.18 \, \text{J} \][/tex]
5. Marble 5:
[tex]\[ KE_5 = \frac{1}{2} \times 0.03 \, \text{kg} \times (3 \, \text{m/s})^2 \][/tex]
[tex]\[ KE_5 = \frac{1}{2} \times 0.03 \times 9 \][/tex]
[tex]\[ KE_5 = 0.5 \times 0.03 \times 9 \][/tex]
[tex]\[ KE_5 = 0.135 \, \text{J} \][/tex]
By comparing the kinetic energy values, we can see that Marble 4, with a kinetic energy of [tex]\( 0.18 \, \text{J} \)[/tex], has the highest kinetic energy.
Therefore, the correct answer is:
D. Marble 4
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