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 given that their kinetic energies are in the ratio 4:1 and they have equal linear momentum.

### Step-by-step Solution:

1. **Understand Kinetic Energy and Momentum**:
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
The linear momentum (p) of an object is given by:
\[ p = mv \]

2. **Express Kinetic Energy in terms of Momentum**:
We can express kinetic energy in terms of momentum. From the momentum formula, we can rearrange it to find velocity:
\[ v = \frac{p}{m} \]
Substituting this into the kinetic energy formula gives:
\[ KE = \frac{1}{2} m \left(\frac{p}{m}\right)^2 = \frac{p^2}{2m} \]

3. **Set up the Ratios**:
Let the kinetic energies of the two bodies be \( KE_1 \) and \( KE_2 \). According to the problem, we have:
\[ \frac{KE_1}{KE_2} = \frac{4}{1} \]
Using the expression for kinetic energy in terms of momentum, we can write:
\[ \frac{KE_1}{KE_2} = \frac{\frac{p^2}{2m_1}}{\frac{p^2}{2m_2}} = \frac{m_2}{m_1} \]

4. **Equate the Ratios**:
From the previous step, we have:
\[ \frac{m_2}{m_1} = \frac{4}{1} \]
This implies:
\[ m_2 = 4m_1 \]

5. **Find the Ratio of Masses**:
To find the ratio \( \frac{m_1}{m_2} \), we can rearrange the equation:
\[ \frac{m_1}{m_2} = \frac{m_1}{4m_1} = \frac{1}{4} \]
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 \( 1 : 4 \).

---