Answer :
Sure! Let's solve this step-by-step:
1. Understanding Kinetic Energy:
The kinetic energy (KE) of an object can be calculated using the formula:
[tex]\[ KE = \frac{1}{2} \times m \times v^2 \][/tex]
where [tex]\(m\)[/tex] is the mass of the object (in kilograms) and [tex]\(v\)[/tex] is the velocity of the object (in meters per second).
2. Convert Mass to Kilograms:
Since the masses of the marbles are given in grams, we need to convert them to kilograms:
- Marble 1: [tex]\(10 \, \text{g} = 0.01 \, \text{kg}\)[/tex]
- Marble 2: [tex]\(20 \, \text{g} = 0.02 \, \text{kg}\)[/tex]
- Marble 3: [tex]\(25 \, \text{g} = 0.025 \, \text{kg}\)[/tex]
- Marble 4: [tex]\(40 \, \text{g} = 0.04 \, \text{kg}\)[/tex]
- Marble 5: [tex]\(30 \, \text{g} = 0.03 \, \text{kg}\)[/tex]
3. Calculating Kinetic Energy for Each Marble:
With the velocity [tex]\(v\)[/tex] given as [tex]\(3 \, \text{m/s}\)[/tex] for all marbles:
- For Marble 1:
[tex]\[ KE_1 = \frac{1}{2} \times 0.01 \, \text{kg} \times (3 \, \text{m/s})^2 = \frac{1}{2} \times 0.01 \times 9 = 0.045 \, \text{J} \][/tex]
- For Marble 2:
[tex]\[ KE_2 = \frac{1}{2} \times 0.02 \, \text{kg} \times (3 \, \text{m/s})^2 = \frac{1}{2} \times 0.02 \times 9 = 0.09 \, \text{J} \][/tex]
- For Marble 3:
[tex]\[ KE_3 = \frac{1}{2} \times 0.025 \, \text{kg} \times (3 \, \text{m/s})^2 = \frac{1}{2} \times 0.025 \times 9 = 0.1125 \, \text{J} \][/tex]
- For Marble 4:
[tex]\[ KE_4 = \frac{1}{2} \times 0.04 \, \text{kg} \times (3 \, \text{m/s})^2 = \frac{1}{2} \times 0.04 \times 9 = 0.18 \, \text{J} \][/tex]
- For Marble 5:
[tex]\[ KE_5 = \frac{1}{2} \times 0.03 \, \text{kg} \times (3 \, \text{m/s})^2 = \frac{1}{2} \times 0.03 \times 9 = 0.135 \, \text{J} \][/tex]
4. Comparing the Kinetic Energies:
Now we compare the calculated kinetic energies:
- Marble 1: [tex]\(0.045 \, \text{J}\)[/tex]
- Marble 2: [tex]\(0.09 \, \text{J}\)[/tex]
- Marble 3: [tex]\(0.1125 \, \text{J}\)[/tex]
- Marble 4: [tex]\(0.18 \, \text{J}\)[/tex]
- Marble 5: [tex]\(0.135 \, \text{J}\)[/tex]
5. Conclusion:
The highest kinetic energy is [tex]\(0.18 \, \text{J}\)[/tex], which belongs to Marble 4.
Therefore, the correct answer is [tex]\( \boxed{D. \, Marble \, 4} \)[/tex].
1. Understanding Kinetic Energy:
The kinetic energy (KE) of an object can be calculated using the formula:
[tex]\[ KE = \frac{1}{2} \times m \times v^2 \][/tex]
where [tex]\(m\)[/tex] is the mass of the object (in kilograms) and [tex]\(v\)[/tex] is the velocity of the object (in meters per second).
2. Convert Mass to Kilograms:
Since the masses of the marbles are given in grams, we need to convert them to kilograms:
- Marble 1: [tex]\(10 \, \text{g} = 0.01 \, \text{kg}\)[/tex]
- Marble 2: [tex]\(20 \, \text{g} = 0.02 \, \text{kg}\)[/tex]
- Marble 3: [tex]\(25 \, \text{g} = 0.025 \, \text{kg}\)[/tex]
- Marble 4: [tex]\(40 \, \text{g} = 0.04 \, \text{kg}\)[/tex]
- Marble 5: [tex]\(30 \, \text{g} = 0.03 \, \text{kg}\)[/tex]
3. Calculating Kinetic Energy for Each Marble:
With the velocity [tex]\(v\)[/tex] given as [tex]\(3 \, \text{m/s}\)[/tex] for all marbles:
- For Marble 1:
[tex]\[ KE_1 = \frac{1}{2} \times 0.01 \, \text{kg} \times (3 \, \text{m/s})^2 = \frac{1}{2} \times 0.01 \times 9 = 0.045 \, \text{J} \][/tex]
- For Marble 2:
[tex]\[ KE_2 = \frac{1}{2} \times 0.02 \, \text{kg} \times (3 \, \text{m/s})^2 = \frac{1}{2} \times 0.02 \times 9 = 0.09 \, \text{J} \][/tex]
- For Marble 3:
[tex]\[ KE_3 = \frac{1}{2} \times 0.025 \, \text{kg} \times (3 \, \text{m/s})^2 = \frac{1}{2} \times 0.025 \times 9 = 0.1125 \, \text{J} \][/tex]
- For Marble 4:
[tex]\[ KE_4 = \frac{1}{2} \times 0.04 \, \text{kg} \times (3 \, \text{m/s})^2 = \frac{1}{2} \times 0.04 \times 9 = 0.18 \, \text{J} \][/tex]
- For Marble 5:
[tex]\[ KE_5 = \frac{1}{2} \times 0.03 \, \text{kg} \times (3 \, \text{m/s})^2 = \frac{1}{2} \times 0.03 \times 9 = 0.135 \, \text{J} \][/tex]
4. Comparing the Kinetic Energies:
Now we compare the calculated kinetic energies:
- Marble 1: [tex]\(0.045 \, \text{J}\)[/tex]
- Marble 2: [tex]\(0.09 \, \text{J}\)[/tex]
- Marble 3: [tex]\(0.1125 \, \text{J}\)[/tex]
- Marble 4: [tex]\(0.18 \, \text{J}\)[/tex]
- Marble 5: [tex]\(0.135 \, \text{J}\)[/tex]
5. Conclusion:
The highest kinetic energy is [tex]\(0.18 \, \text{J}\)[/tex], which belongs to Marble 4.
Therefore, the correct answer is [tex]\( \boxed{D. \, Marble \, 4} \)[/tex].
