If the radii of circular path of two particles are in the ratio of `1 :2`, then in order to have same centripatal acceleration, their speeds should be in the ratio of :

Options

A. `1:4`
B. `4:1`
C. `1:sqrt(2)`
D. `sqrt(2):1`

Explanation

To solve the problem, we need to find the ratio of the speeds of two particles moving in circular paths with given radii, while ensuring they have the same centripetal acceleration.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The radii of the circular paths of two particles are in the ratio \( r_1 : r_2 = 1 : 2 \).
- This means we can express the radii as \( r_1 = r \) and \( r_2 = 2r \) for some radius \( r \).

2. **Centripetal Acceleration Formula:**
- The formula for centripetal acceleration \( a_c \) is given by:
\[ a_c = \frac{v^2}{r} \]
- Where \( v \) is the speed of the particle and \( r \) is the radius of the circular path.

3. **Set Up the Equation for Both Particles:**
- For the first particle:
\[ a_{c1} = \frac{v_1^2}{r_1} = \frac{v_1^2}{r} \]
- For the second particle:
\[ a_{c2} = \frac{v_2^2}{r_2} = \frac{v_2^2}{2r} \]

4. **Equate the Centripetal Accelerations:**
- Since both particles have the same centripetal acceleration:
\[ \frac{v_1^2}{r} = \frac{v_2^2}{2r} \]

5. **Simplify the Equation:**
- We can cancel \( r \) from both sides (assuming \( r \neq 0 \)):
\[ v_1^2 = \frac{v_2^2}{2} \]

6. **Express the Speeds in Ratio Form:**
- Rearranging gives:
\[ \frac{v_1^2}{v_2^2} = \frac{1}{2} \]
- Taking the square root of both sides:
\[ \frac{v_1}{v_2} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

7. **Final Ratio of Speeds:**
- Therefore, the ratio of the speeds \( v_1 : v_2 \) is:
\[ v_1 : v_2 = 1 : \sqrt{2} \]

### Conclusion:
The speeds of the two particles should be in the ratio of \( 1 : \sqrt{2} \).