Dive into the world of knowledge and get your queries resolved at IDNLearn.com. Discover comprehensive answers to your questions from our community of knowledgeable experts.

Select the correct answer.

Five marbles roll down a ramp. Each marble reaches the bottom of the ramp at a speed of 3 meters/second. Which marble has the highest kinetic energy at the bottom of the ramp? The table lists the mass of each marble:

\begin{tabular}{|c|c|c|c|c|}
\hline Marble 1 & Marble 2 & Marble 3 & Marble 4 & Marble 5 \\
\hline [tex]$10 g$[/tex] & [tex]$20 g$[/tex] & [tex]$25 g$[/tex] & [tex]$40 g$[/tex] & [tex]$30 g$[/tex] \\
\hline
\end{tabular}

A. Marble 1
B. Marble 2
C. Marble 3
D. Marble 4
E. Marble 5


Sagot :

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].