Two bodies of different masses `m_(1)` and `m_(2)` have equal momenta. Their kinetic energies `E_(1)` and `E_(2)` are in the ratio

Options

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

Explanation

To solve the problem, we need to find the ratio of the kinetic energies \( E_1 \) and \( E_2 \) of two bodies with different masses \( m_1 \) and \( m_2 \) that have equal momenta.

### Step-by-Step Solution:

1. **Understand the relationship between momentum and kinetic energy**:
The momentum \( p \) of an object is given by:
\[ p = mv \]
where \( m \) is the mass and \( v \) is the velocity of the object.

2. **Write the expression for kinetic energy**:
The kinetic energy \( E \) of an object is given by:
\[ E = \frac{1}{2} mv^2 \]

3. **Express kinetic energy in terms of momentum**:
We can express kinetic energy using momentum:
\[ E = \frac{p^2}{2m} \]
Thus, for the two bodies, we can write:
\[ E_1 = \frac{p^2}{2m_1} \quad \text{and} \quad E_2 = \frac{p^2}{2m_2} \]

4. **Set the momenta equal**:
Since both bodies have equal momentum, we can denote this common momentum as \( p \).

5. **Substitute the expressions for kinetic energy**:
Now, substituting the expressions for kinetic energies:
\[ E_1 = \frac{p^2}{2m_1} \quad \text{and} \quad E_2 = \frac{p^2}{2m_2} \]

6. **Find the ratio of the kinetic energies**:
To find the ratio \( \frac{E_1}{E_2} \):
\[ \frac{E_1}{E_2} = \frac{\frac{p^2}{2m_1}}{\frac{p^2}{2m_2}} = \frac{p^2}{2m_1} \cdot \frac{2m_2}{p^2} = \frac{m_2}{m_1} \]

7. **Final result**:
Therefore, the ratio of the kinetic energies \( E_1 \) and \( E_2 \) is:
\[ \frac{E_1}{E_2} = \frac{m_2}{m_1} \]

### Conclusion:
The ratio of the kinetic energies \( E_1 \) and \( E_2 \) is \( \frac{m_2}{m_1} \).