Answer :
To determine which marble has the highest kinetic energy at the bottom of the ramp, we need to calculate the kinetic energy for each marble. The kinetic energy (KE) can be calculated using the formula:
[tex]\[ KE = \frac{1}{2}mv^2 \][/tex]
where
- \( m \) is the mass of the marble in kilograms (kg),
- \( v \) is the speed of the marble in meters per second (m/s), which is given as 3 m/s for all marbles.
First, let's convert the mass of each marble from grams to kilograms because the standard unit for mass in physics calculations is kilograms.
[tex]\[ \text{Mass in kilograms} = \text{Mass in grams} / 1000 \][/tex]
Here are the masses of the marbles in kilograms:
- Marble 1: \( 10 \, \text{g} = 10 / 1000 = 0.01 \, \text{kg} \)
- Marble 2: \( 20 \, \text{g} = 20 / 1000 = 0.02 \, \text{kg} \)
- Marble 3: \( 25 \, \text{g} = 25 / 1000 = 0.025 \, \text{kg} \)
- Marble 4: \( 40 \, \text{g} = 40 / 1000 = 0.04 \, \text{kg} \)
- Marble 5: \( 30 \, \text{g} = 30 / 1000 = 0.03 \, \text{kg} \)
Next, we calculate the kinetic energy for each marble:
- For Marble 1:
[tex]\[ KE_1 = \frac{1}{2} \times 0.01 \, \text{kg} \times (3 \, \text{m/s})^2 = 0.5 \times 0.01 \times 9 = 0.045 \, \text{Joules} \][/tex]
- For Marble 2:
[tex]\[ KE_2 = \frac{1}{2} \times 0.02 \, \text{kg} \times (3 \, \text{m/s})^2 = 0.5 \times 0.02 \times 9 = 0.09 \, \text{Joules} \][/tex]
- For Marble 3:
[tex]\[ KE_3 = \frac{1}{2} \times 0.025 \, \text{kg} \times (3 \, \text{m/s})^2 = 0.5 \times 0.025 \times 9 = 0.1125 \, \text{Joules} \][/tex]
- For Marble 4:
[tex]\[ KE_4 = \frac{1}{2} \times 0.04 \, \text{kg} \times (3 \, \text{m/s})^2 = 0.5 \times 0.04 \times 9 = 0.18 \, \text{Joules} \][/tex]
- For Marble 5:
[tex]\[ KE_5 = \frac{1}{2} \times 0.03 \, \text{kg} \times (3 \, \text{m/s})^2 = 0.5 \times 0.03 \times 9 = 0.135 \, \text{Joules} \][/tex]
Comparing these kinetic energies:
- Marble 1: \( 0.045 \, \text{J} \)
- Marble 2: \( 0.09 \, \text{J} \)
- Marble 3: \( 0.1125 \, \text{J} \)
- Marble 4: \( 0.18 \, \text{J} \)
- Marble 5: \( 0.135 \, \text{J} \)
The marble with the highest kinetic energy is Marble 4 with \( 0.18 \, \text{J} \).
Therefore, the correct answer is:
D. Marble 4
[tex]\[ KE = \frac{1}{2}mv^2 \][/tex]
where
- \( m \) is the mass of the marble in kilograms (kg),
- \( v \) is the speed of the marble in meters per second (m/s), which is given as 3 m/s for all marbles.
First, let's convert the mass of each marble from grams to kilograms because the standard unit for mass in physics calculations is kilograms.
[tex]\[ \text{Mass in kilograms} = \text{Mass in grams} / 1000 \][/tex]
Here are the masses of the marbles in kilograms:
- Marble 1: \( 10 \, \text{g} = 10 / 1000 = 0.01 \, \text{kg} \)
- Marble 2: \( 20 \, \text{g} = 20 / 1000 = 0.02 \, \text{kg} \)
- Marble 3: \( 25 \, \text{g} = 25 / 1000 = 0.025 \, \text{kg} \)
- Marble 4: \( 40 \, \text{g} = 40 / 1000 = 0.04 \, \text{kg} \)
- Marble 5: \( 30 \, \text{g} = 30 / 1000 = 0.03 \, \text{kg} \)
Next, we calculate the kinetic energy for each marble:
- For Marble 1:
[tex]\[ KE_1 = \frac{1}{2} \times 0.01 \, \text{kg} \times (3 \, \text{m/s})^2 = 0.5 \times 0.01 \times 9 = 0.045 \, \text{Joules} \][/tex]
- For Marble 2:
[tex]\[ KE_2 = \frac{1}{2} \times 0.02 \, \text{kg} \times (3 \, \text{m/s})^2 = 0.5 \times 0.02 \times 9 = 0.09 \, \text{Joules} \][/tex]
- For Marble 3:
[tex]\[ KE_3 = \frac{1}{2} \times 0.025 \, \text{kg} \times (3 \, \text{m/s})^2 = 0.5 \times 0.025 \times 9 = 0.1125 \, \text{Joules} \][/tex]
- For Marble 4:
[tex]\[ KE_4 = \frac{1}{2} \times 0.04 \, \text{kg} \times (3 \, \text{m/s})^2 = 0.5 \times 0.04 \times 9 = 0.18 \, \text{Joules} \][/tex]
- For Marble 5:
[tex]\[ KE_5 = \frac{1}{2} \times 0.03 \, \text{kg} \times (3 \, \text{m/s})^2 = 0.5 \times 0.03 \times 9 = 0.135 \, \text{Joules} \][/tex]
Comparing these kinetic energies:
- Marble 1: \( 0.045 \, \text{J} \)
- Marble 2: \( 0.09 \, \text{J} \)
- Marble 3: \( 0.1125 \, \text{J} \)
- Marble 4: \( 0.18 \, \text{J} \)
- Marble 5: \( 0.135 \, \text{J} \)
The marble with the highest kinetic energy is Marble 4 with \( 0.18 \, \text{J} \).
Therefore, the correct answer is:
D. Marble 4
