Two bodies of masses `m_(1)` and `m_(2)` have equal `KE`. Their momenta is in the ratio

Answer

To solve the problem, we need to find the ratio of momenta of two bodies with equal kinetic energy. Let's denote the masses of the two bodies as \( m_1 \) and \( m_2 \), and their velocities as \( v_1 \) and \( v_2 \) respectively.

### Step-by-Step Solution:

1. **Understand the Kinetic Energy Condition**:
Since both bodies have equal kinetic energy, we can write:
\[ KE_1 = KE_2 \]
This translates to:
\[ \frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_2 v_2^2 \]

2. **Simplify the Equation**:
The \( \frac{1}{2} \) cancels out from both sides:
\[ m_1 v_1^2 = m_2 v_2^2 \]

3. **Rearranging the Equation**:
We can rearrange this equation to express the relationship between the squares of the velocities:
\[ \frac{v_1^2}{v_2^2} = \frac{m_2}{m_1} \]

4. **Taking the Square Root**:
Taking the square root of both sides gives us the ratio of the velocities:
\[ \frac{v_1}{v_2} = \sqrt{\frac{m_2}{m_1}} \]

5. **Finding the Ratio of Momenta**:
The momentum \( p \) of an object is given by the product of its mass and velocity:
\[ p_1 = m_1 v_1 \quad \text{and} \quad p_2 = m_2 v_2 \]
Therefore, the ratio of the momenta is:
\[ \frac{p_1}{p_2} = \frac{m_1 v_1}{m_2 v_2} \]

6. **Substituting the Velocity Ratio**:
Substituting \( \frac{v_1}{v_2} = \sqrt{\frac{m_2}{m_1}} \) into the momentum ratio gives:
\[ \frac{p_1}{p_2} = \frac{m_1}{m_2} \cdot \sqrt{\frac{m_2}{m_1}} \]

7. **Simplifying the Expression**:
This can be simplified as follows:
\[ \frac{p_1}{p_2} = \frac{m_1}{m_2} \cdot \frac{\sqrt{m_2}}{\sqrt{m_1}} = \frac{m_1^{1/2}}{m_2^{1/2}} = \sqrt{\frac{m_1}{m_2}} \]

8. **Final Result**:
Thus, the ratio of the momenta of the two bodies is:
\[ \frac{p_1}{p_2} = \sqrt{\frac{m_1}{m_2}} \]

### Conclusion:
The ratio of the momenta of the two bodies is \( \sqrt{\frac{m_1}{m_2}} \).