Two bodies with kinetic energies in the ratio 4:1 are moving with equal linear momentum. The ratio of their masses is

Options

A. `1:2`
B. `1:1`
C. `4:1`
D. `1:4`

Answer

To solve the problem, we need to find the ratio of the masses of two bodies that have kinetic energies in the ratio of 4:1 and are moving with equal linear momentum.

### Step-by-Step Solution:

1. **Understanding Kinetic Energy and Momentum**:
- The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{p^2}{2m} \]
where \( p \) is the linear momentum and \( m \) is the mass of the object.
- Since both bodies have equal linear momentum, we can denote their momenta as \( p \).

2. **Setting Up the Ratios**:
- Let the kinetic energies of the two bodies be \( KE_1 \) and \( KE_2 \).
- According to the problem, the ratio of their kinetic energies is:
\[ \frac{KE_1}{KE_2} = \frac{4}{1} \]

3. **Expressing Kinetic Energies in Terms of Mass**:
- For the first body:
\[ KE_1 = \frac{p^2}{2m_1} \]
- For the second body:
\[ KE_2 = \frac{p^2}{2m_2} \]

4. **Setting Up the Equation**:
- Now, substituting the expressions for kinetic energies into the ratio:
\[ \frac{KE_1}{KE_2} = \frac{\frac{p^2}{2m_1}}{\frac{p^2}{2m_2}} = \frac{m_2}{m_1} \]
- Since \( p^2 \) and \( 2 \) are common in both the numerator and denominator, they cancel out.

5. **Using the Given Ratio**:
- We know from the problem that:
\[ \frac{KE_1}{KE_2} = \frac{4}{1} \]
- Therefore, we can equate this to our previous expression:
\[ \frac{m_2}{m_1} = \frac{4}{1} \]

6. **Finding the Mass Ratio**:
- Rearranging gives us:
\[ \frac{m_1}{m_2} = \frac{1}{4} \]
- Thus, the ratio of the masses \( m_1:m_2 \) is:
\[ m_1:m_2 = 1:4 \]

### Final Answer:
The ratio of their masses is \( 1:4 \).