Two bodies of masses `m_(1)` and `m_(2)`, respectively, are thrown vertically upwards with equal initial linear momenta. Then the ratio of maximum heights attained by the first to that by the second body is

Options

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

Answer

To solve the problem, we need to determine the ratio of the maximum heights attained by two bodies of different masses when they are thrown upwards with equal initial linear momenta.

### Step-by-Step Solution:

1. **Understanding Initial Momentum**:
Given that the two bodies have equal initial linear momenta, we can express this mathematically:
\[ m_1 v_1 = m_2 v_2 \]
where \( m_1 \) and \( m_2 \) are the masses of the first and second body, respectively, and \( v_1 \) and \( v_2 \) are their respective initial velocities.

2. **Expressing the Ratio of Velocities**:
From the momentum equation, we can derive the ratio of their velocities:
\[ \frac{v_1}{v_2} = \frac{m_2}{m_1} \]

3. **Using the Equation of Motion**:
We will use the equation of motion to find the maximum height attained by each body. The equation we will use is:
\[ v^2 - u^2 = 2as \]
where \( v \) is the final velocity (0 at maximum height), \( u \) is the initial velocity, \( a \) is the acceleration (which is \(-g\), the acceleration due to gravity), and \( s \) is the height.

4. **For the First Body**:
For the first body, we have:
\[ 0 - v_1^2 = 2(-g) S_1 \]
Rearranging gives:
\[ S_1 = \frac{v_1^2}{2g} \]

5. **For the Second Body**:
Similarly, for the second body:
\[ 0 - v_2^2 = 2(-g) S_2 \]
Rearranging gives:
\[ S_2 = \frac{v_2^2}{2g} \]

6. **Finding the Ratio of Heights**:
Now we can find the ratio of the heights \( S_1 \) and \( S_2 \):
\[ \frac{S_1}{S_2} = \frac{\frac{v_1^2}{2g}}{\frac{v_2^2}{2g}} = \frac{v_1^2}{v_2^2} \]

7. **Substituting the Velocity Ratio**:
We already found that \( \frac{v_1}{v_2} = \frac{m_2}{m_1} \). Squaring this gives:
\[ \frac{v_1^2}{v_2^2} = \left(\frac{m_2}{m_1}\right)^2 \]

8. **Final Ratio of Heights**:
Therefore, substituting back into the height ratio:
\[ \frac{S_1}{S_2} = \left(\frac{m_2}{m_1}\right)^2 \]

### Conclusion:
The ratio of the maximum heights attained by the first body to that of the second body is:
\[ \frac{S_1}{S_2} = \frac{m_2^2}{m_1^2} \]

### Answer:
The correct option is \( \frac{m_2^2}{m_1^2} \).