Two particles of equal masses have ratio of their linear momenta as `2:3`. The ratio of kinetic energies will be

Options

A. `4: 9`
B. `2:3`
C. `sqrt2: sqrt3`
D. `3:2`

Explanation

To solve the problem, we need to find the ratio of the kinetic energies of two particles given that their linear momentum ratio is 2:3 and that they have equal masses.

### Step-by-Step Solution:

1. **Understanding Linear Momentum**:
- The linear momentum \( P \) of an object is given by the formula:
\[ P = m \cdot v \]
- Where \( m \) is the mass and \( v \) is the velocity of the object.

2. **Given Information**:
- Let the mass of each particle be \( m \).
- The ratio of their linear momenta is given as:
\[ \frac{P_1}{P_2} = \frac{2}{3} \]

3. **Expressing Linear Momentum in Terms of Velocity**:
- For the first particle:
\[ P_1 = m \cdot v_1 \]
- For the second particle:
\[ P_2 = m \cdot v_2 \]
- Therefore, the ratio of their momenta can be expressed as:
\[ \frac{m \cdot v_1}{m \cdot v_2} = \frac{v_1}{v_2} = \frac{2}{3} \]

4. **Finding the Ratio of Velocities**:
- From the above equation, we can conclude:
\[ \frac{v_1}{v_2} = \frac{2}{3} \]

5. **Kinetic Energy Formula**:
- The kinetic energy \( K \) of an object is given by:
\[ K = \frac{1}{2} m v^2 \]
- For the first particle:
\[ K_1 = \frac{1}{2} m v_1^2 \]
- For the second particle:
\[ K_2 = \frac{1}{2} m v_2^2 \]

6. **Finding the Ratio of Kinetic Energies**:
- The ratio of the kinetic energies can be expressed as:
\[ \frac{K_1}{K_2} = \frac{\frac{1}{2} m v_1^2}{\frac{1}{2} m v_2^2} = \frac{v_1^2}{v_2^2} \]
- Since \( \frac{v_1}{v_2} = \frac{2}{3} \), we can square this ratio:
\[ \frac{K_1}{K_2} = \left(\frac{v_1}{v_2}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]

7. **Final Result**:
- Therefore, the ratio of the kinetic energies of the two particles is:
\[ \frac{K_1}{K_2} = \frac{4}{9} \]

### Conclusion:
The ratio of the kinetic energies of the two particles is \( \frac{4}{9} \).