Two bodies of masses `m_(1)` and `m_(2)` are acted upon by a constant force `F` for a time `t` . They start from rest and acquire kinetic energies `K_(1)` and `K_(2)` respectively. Then `(K_(1))/(K_(2))` is `:`

Options

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

Explanation

To solve the problem, we need to find the ratio of the kinetic energies \( K_1 \) and \( K_2 \) of two bodies with masses \( m_1 \) and \( m_2 \) that are acted upon by a constant force \( F \) for a time \( t \).

### Step-by-Step Solution:

1. **Identify the acceleration of each body**:
- The acceleration \( a_1 \) of the first body (mass \( m_1 \)) is given by:
\[ a_1 = \frac{F}{m_1} \]
- The acceleration \( a_2 \) of the second body (mass \( m_2 \)) is given by:
\[ a_2 = \frac{F}{m_2} \]

**Hint**: Use Newton's second law \( F = ma \) to find the acceleration for each mass.

2. **Calculate the final velocity of each body**:
- The final velocity \( v_1 \) of the first body after time \( t \) is:
\[ v_1 = a_1 \cdot t = \frac{F}{m_1} \cdot t = \frac{Ft}{m_1} \]
- The final velocity \( v_2 \) of the second body after time \( t \) is:
\[ v_2 = a_2 \cdot t = \frac{F}{m_2} \cdot t = \frac{Ft}{m_2} \]

**Hint**: Remember that the bodies start from rest, so initial velocity is zero.

3. **Determine the kinetic energy of each body**:
- The kinetic energy \( K_1 \) of the first body is given by:
\[ K_1 = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_1 \left(\frac{Ft}{m_1}\right)^2 = \frac{1}{2} m_1 \cdot \frac{F^2 t^2}{m_1^2} = \frac{F^2 t^2}{2 m_1} \]
- The kinetic energy \( K_2 \) of the second body is given by:
\[ K_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} m_2 \left(\frac{Ft}{m_2}\right)^2 = \frac{1}{2} m_2 \cdot \frac{F^2 t^2}{m_2^2} = \frac{F^2 t^2}{2 m_2} \]

**Hint**: Use the formula for kinetic energy \( K = \frac{1}{2} mv^2 \) to derive the kinetic energies.

4. **Find the ratio of the kinetic energies**:
- The ratio \( \frac{K_1}{K_2} \) is:
\[ \frac{K_1}{K_2} = \frac{\frac{F^2 t^2}{2 m_1}}{\frac{F^2 t^2}{2 m_2}} = \frac{m_2}{m_1} \]

**Hint**: Simplify the fraction by canceling out common terms.

5. **Conclusion**:
- Therefore, the ratio of the kinetic energies \( \frac{K_1}{K_2} \) is:
\[ \frac{K_1}{K_2} = \frac{m_2}{m_1} \]

### Final Answer:
The ratio \( \frac{K_1}{K_2} \) is \( \frac{m_2}{m_1} \).