Two balls of masses 2 g and 6 g are moving with kinetic energy in the ratio of `3:1`. What is the ratio of their linear momentum ?

Options

A. `1:1`
B. `2:1`
C. `1:2`
D. None of these

Answer

To solve the problem, we need to find the ratio of the linear momentum of two balls given their masses and the ratio of their kinetic energies.

### Step-by-Step Solution:

1. **Identify the given data:**
- Mass of ball 1, \( m_1 = 2 \, \text{g} \)
- Mass of ball 2, \( m_2 = 6 \, \text{g} \)
- Ratio of their kinetic energies, \( KE_1 : KE_2 = 3 : 1 \)

2. **Recall the formula for kinetic energy:**
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass and \( v \) is the velocity of the object.

3. **Express the kinetic energies in terms of momentum:**
The linear momentum \( p \) is defined as:
\[ p = mv \]
From this, we can express the velocity in terms of momentum:
\[ v = \frac{p}{m} \]
Substituting this into the kinetic energy formula gives:
\[ KE = \frac{1}{2} m \left(\frac{p}{m}\right)^2 = \frac{p^2}{2m} \]

4. **Set up the ratio of kinetic energies:**
Using the expression for kinetic energy in terms of momentum, we can write:
\[ \frac{KE_1}{KE_2} = \frac{\frac{p_1^2}{2m_1}}{\frac{p_2^2}{2m_2}} = \frac{p_1^2 \cdot m_2}{p_2^2 \cdot m_1} \]
Given that \( KE_1 : KE_2 = 3 : 1 \), we can write:
\[ \frac{p_1^2 \cdot m_2}{p_2^2 \cdot m_1} = 3 \]

5. **Substitute the masses:**
Substitute \( m_1 = 2 \, \text{g} \) and \( m_2 = 6 \, \text{g} \):
\[ \frac{p_1^2 \cdot 6}{p_2^2 \cdot 2} = 3 \]

6. **Simplify the equation:**
Rearranging gives:
\[ \frac{p_1^2}{p_2^2} = \frac{3 \cdot 2}{6} = 1 \]
This implies:
\[ p_1^2 = p_2^2 \]

7. **Take the square root:**
Taking the square root of both sides results in:
\[ p_1 = p_2 \]

8. **Conclusion:**
The ratio of their linear momentum is:
\[ \frac{p_1}{p_2} = 1 : 1 \]

### Final Answer:
The ratio of their linear momentum is \( 1 : 1 \).