Two bodies are having kinetic energies in the radio `16:9` If they have same linear momentum , the ratio of their masses respectively is :

Options

A. `4:3`
B. `3:4`
C. `16:9`
D. `9:16`

Answer

To solve the problem, we need to establish the relationship between the kinetic energies, linear momentum, and masses of the two bodies.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- Let the kinetic energies of the two bodies be \( K_1 \) and \( K_2 \).
- We are given that the ratio of their kinetic energies is \( \frac{K_1}{K_2} = \frac{16}{9} \).
- Both bodies have the same linear momentum \( p \).

2. **Using the Kinetic Energy Formula**:
- The formula for kinetic energy is given by:
\[ K = \frac{p^2}{2m} \]
- For body 1, we can write:
\[ K_1 = \frac{p^2}{2m_1} \]
- For body 2, we can write:
\[ K_2 = \frac{p^2}{2m_2} \]

3. **Setting Up the Ratio of Kinetic Energies**:
- Now, we can find the ratio of the kinetic energies:
\[ \frac{K_1}{K_2} = \frac{\frac{p^2}{2m_1}}{\frac{p^2}{2m_2}} = \frac{m_2}{m_1} \]
- Since \( p^2 \) and \( 2 \) are constants, they cancel out.

4. **Substituting the Given Ratio**:
- From the problem, we have:
\[ \frac{K_1}{K_2} = \frac{16}{9} \]
- Therefore, we can equate:
\[ \frac{m_2}{m_1} = \frac{16}{9} \]

5. **Finding the Ratio of Masses**:
- Rearranging the above equation gives:
\[ \frac{m_1}{m_2} = \frac{9}{16} \]
- Thus, the ratio of their masses is \( m_1:m_2 = 9:16 \).

### Final Answer:
The ratio of their masses is \( 9:16 \).

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