Two moving bodies having mass in the ratio `1: 2` and kinetic energy in the ratio `1:8`. Find the ratio of their velocities.

Options

A. `1:6`
B. `1:2`
C. `1:4`
D. `1:8`

Explanation

To solve the problem, we need to find the ratio of the velocities of two moving bodies given their mass and kinetic energy ratios.

### Step-by-Step Solution:

1. **Identify the Given Ratios:**
- Mass ratio: \( \frac{m_1}{m_2} = \frac{1}{2} \)
- Kinetic energy ratio: \( \frac{K_1}{K_2} = \frac{1}{8} \)

2. **Write the Formula for Kinetic Energy:**
The kinetic energy (K.E.) of a body is given by the formula:
\[ K = \frac{1}{2} mv^2 \]
For two bodies, we can write:
\[ K_1 = \frac{1}{2} m_1 v_1^2 \quad \text{and} \quad K_2 = \frac{1}{2} m_2 v_2^2 \]

3. **Set Up the Kinetic Energy Ratio:**
Using the kinetic energy expressions, we can set up the ratio:
\[ \frac{K_1}{K_2} = \frac{\frac{1}{2} m_1 v_1^2}{\frac{1}{2} m_2 v_2^2} = \frac{m_1 v_1^2}{m_2 v_2^2} \]
Given that \( \frac{K_1}{K_2} = \frac{1}{8} \), we have:
\[ \frac{m_1 v_1^2}{m_2 v_2^2} = \frac{1}{8} \]

4. **Substitute the Mass Ratio:**
From the mass ratio \( \frac{m_1}{m_2} = \frac{1}{2} \), we can express \( m_1 \) in terms of \( m_2 \):
\[ m_1 = \frac{1}{2} m_2 \]
Substituting this into the kinetic energy ratio gives:
\[ \frac{\frac{1}{2} m_2 v_1^2}{m_2 v_2^2} = \frac{1}{8} \]
This simplifies to:
\[ \frac{v_1^2}{2 v_2^2} = \frac{1}{8} \]

5. **Cross-Multiply to Solve for Velocity Ratio:**
Cross-multiplying gives:
\[ 8 v_1^2 = 2 v_2^2 \]
Dividing both sides by 2:
\[ 4 v_1^2 = v_2^2 \]

6. **Take the Square Root:**
Taking the square root of both sides yields:
\[ \frac{v_1}{v_2} = \frac{1}{2} \]

7. **Final Ratio of Velocities:**
Therefore, the ratio of the velocities \( v_1 : v_2 \) is:
\[ v_1 : v_2 = 1 : 2 \]

### Conclusion:
The ratio of the velocities of the two bodies is \( 1:2 \).