To determine which marble has the highest kinetic energy at the bottom of the ramp, we need to consider the kinetic energy formula:
[tex]\[ \text{KE} = \frac{1}{2} m v^2 \][/tex]
where:
- [tex]\( \text{KE} \)[/tex] is the kinetic energy,
- [tex]\( m \)[/tex] is the mass in kilograms,
- [tex]\( v \)[/tex] is the velocity in meters per second.
Given the masses of the marbles and their speed of 3 meters/second, we can determine the kinetic energy for each marble.
1. Convert the masses from grams to kilograms:
[tex]\[
\text{Marble 1: } 10\,g = 0.01\,kg \\
\text{Marble 2: } 20\,g = 0.02\,kg \\
\text{Marble 3: } 25\,g = 0.025\,kg \\
\text{Marble 4: } 40\,g = 0.04\,kg \\
\text{Marble 5: } 30\,g = 0.03\,kg
\][/tex]
2. Calculate the kinetic energy for each marble:
[tex]\[
\text{KE}_1 = \frac{1}{2} \times 0.01\,kg \times (3\,m/s)^2 = \frac{1}{2} \times 0.01 \times 9 = 0.045\,J \\
\text{KE}_2 = \frac{1}{2} \times 0.02\,kg \times (3\,m/s)^2 = \frac{1}{2} \times 0.02 \times 9 = 0.09\,J \\
\text{KE}_3 = \frac{1}{2} \times 0.025\,kg \times (3\,m/s)^2 = \frac{1}{2} \times 0.025 \times 9 = 0.1125\,J \\
\text{KE}_4 = \frac{1}{2} \times 0.04\,kg \times (3\,m/s)^2 = \frac{1}{2} \times 0.04 \times 9 = 0.18\,J \\
\text{KE}_5 = \frac{1}{2} \times 0.03\,kg \times (3\,m/s)^2 = \frac{1}{2} \times 0.03 \times 9 = 0.135\,J
\][/tex]
3. Compare the kinetic energies:
[tex]\[
\text{Marble 1: } 0.045\,J \\
\text{Marble 2: } 0.09\,J \\
\text{Marble 3: } 0.1125\,J \\
\text{Marble 4: } 0.18\,J \\
\text{Marble 5: } 0.135\,J
\][/tex]
From the calculations, it’s clear that Marble 4 has the highest kinetic energy of [tex]\( 0.18\,J \)[/tex].
Therefore, the marble with the highest kinetic energy at the bottom of the ramp is:
D. Marble 4
