Two bodies with masses `m1 kg and m2 kg` have equal kinetic energies. If `p_(1) and P_(2)` are their respective momenta, then `p_(1)//P_(2)` is equal to :

Options

A. `(m_(1))/(m_(2))`
B. `sqrt((m_(2))/(m_(1)))`
C. `sqrt((m_(1))/(m_(2)))`
D. `(m_(1)^(2))/(m_(2)^(2))`

Answer

To solve the problem, we need to analyze the relationship between kinetic energy and momentum for two bodies with masses \( m_1 \) and \( m_2 \) that have equal kinetic energies.

### Step-by-Step Solution:

1. **Understanding Kinetic Energy**:
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass and \( v \) is the velocity of the object.

2. **Setting Up the Equation for Equal Kinetic Energies**:
Since both bodies have equal kinetic energies, we can write:
\[ KE_1 = KE_2 \]
This gives us:
\[ \frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_2 v_2^2 \]
Simplifying this, we get:
\[ m_1 v_1^2 = m_2 v_2^2 \]

3. **Expressing Momentum**:
The momentum \( p \) of an object is given by:
\[ p = mv \]
Therefore, for the two bodies, we have:
\[ p_1 = m_1 v_1 \quad \text{and} \quad p_2 = m_2 v_2 \]

4. **Relating Momentum to Kinetic Energy**:
From the kinetic energy equation, we can express \( v_1 \) and \( v_2 \) in terms of kinetic energy \( KE \):
\[ v_1 = \sqrt{\frac{2 KE}{m_1}} \quad \text{and} \quad v_2 = \sqrt{\frac{2 KE}{m_2}} \]

5. **Substituting for Momentum**:
Now substituting \( v_1 \) and \( v_2 \) into the momentum equations:
\[ p_1 = m_1 \sqrt{\frac{2 KE}{m_1}} = \sqrt{2 m_1 KE} \]
\[ p_2 = m_2 \sqrt{\frac{2 KE}{m_2}} = \sqrt{2 m_2 KE} \]

6. **Finding the Ratio of Momenta**:
Now, we can find the ratio of the momenta:
\[ \frac{p_1}{p_2} = \frac{\sqrt{2 m_1 KE}}{\sqrt{2 m_2 KE}} = \frac{\sqrt{m_1}}{\sqrt{m_2}} = \sqrt{\frac{m_1}{m_2}} \]

7. **Conclusion**:
Therefore, the ratio of the momenta \( \frac{p_1}{p_2} \) is equal to:
\[ \frac{p_1}{p_2} = \sqrt{\frac{m_1}{m_2}} \]

### Final Answer:
\[ \frac{p_1}{p_2} = \sqrt{\frac{m_1}{m_2}} \]