Two bodies with kinetic energies in the ratio 2 : 3 are moving with equal momentum . If `m_(1) = x m_(2)` , then the value of x is _______

Explanation

To solve the problem, we need to analyze the relationship between kinetic energy, momentum, and mass for two bodies.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We know that the kinetic energies of two bodies (K1 and K2) are in the ratio of 2:3, and they are moving with equal momentum (P).

2. **Using the Formula for Kinetic Energy**:
The kinetic energy (K) of an object is given by the formula:
\[ K = \frac{p^2}{2m} \]
where \( p \) is the momentum and \( m \) is the mass of the object.

3. **Setting Up the Equations**:
Since both bodies have the same momentum (P), we can express their kinetic energies as:
\[ K_1 = \frac{P^2}{2m_1} \quad \text{and} \quad K_2 = \frac{P^2}{2m_2} \]

4. **Using the Ratio of Kinetic Energies**:
We know from the problem that:
\[ \frac{K_1}{K_2} = \frac{2}{3} \]
Substituting the expressions for K1 and K2, we get:
\[ \frac{\frac{P^2}{2m_1}}{\frac{P^2}{2m_2}} = \frac{2}{3} \]

5. **Simplifying the Equation**:
The \( P^2 \) and \( 2 \) cancel out, leading to:
\[ \frac{m_2}{m_1} = \frac{2}{3} \]

6. **Finding the Ratio of Masses**:
Rearranging the above equation gives:
\[ \frac{m_1}{m_2} = \frac{3}{2} \]
Therefore, we can express this as:
\[ m_1 = \frac{3}{2} m_2 \]

7. **Expressing in Terms of x**:
Since \( m_1 = x m_2 \), we can equate:
\[ x = \frac{3}{2} \]

### Final Answer:
Thus, the value of \( x \) is \( \frac{3}{2} \).

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