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 with given masses and kinetic energy ratios.

### Step-by-Step Solution:

1. **Understanding 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.

2. **Setting Up the Problem**:
We have two balls with masses:
- Mass of ball 1, \( m_1 = 2 \, \text{g} \)
- Mass of ball 2, \( m_2 = 6 \, \text{g} \)

The kinetic energy ratio is given as:
\[ \frac{KE_1}{KE_2} = \frac{3}{1} \]

3. **Expressing Kinetic Energies**:
Using the kinetic energy formula, we can express the kinetic energies of the two balls:
\[ KE_1 = \frac{1}{2} m_1 v_1^2 \]
\[ KE_2 = \frac{1}{2} m_2 v_2^2 \]

4. **Setting Up the Ratio**:
Now, we can write the ratio of their kinetic energies:
\[ \frac{KE_1}{KE_2} = \frac{\frac{1}{2} m_1 v_1^2}{\frac{1}{2} m_2 v_2^2} = \frac{m_1 v_1^2}{m_2 v_2^2} \]
Substituting the given ratio:
\[ \frac{m_1 v_1^2}{m_2 v_2^2} = \frac{3}{1} \]

5. **Substituting Masses**:
Substitute \( m_1 = 2 \, \text{g} \) and \( m_2 = 6 \, \text{g} \):
\[ \frac{2 v_1^2}{6 v_2^2} = 3 \]

6. **Simplifying the Equation**:
Simplifying the equation gives:
\[ \frac{v_1^2}{v_2^2} = \frac{3 \times 6}{2} = 9 \]
Therefore, we have:
\[ \frac{v_1}{v_2} = 3 \]

7. **Calculating Momentum**:
The linear momentum \( P \) is given by:
\[ P = mv \]
Therefore, the momenta of the two balls are:
\[ P_1 = m_1 v_1 = 2 v_1 \]
\[ P_2 = m_2 v_2 = 6 v_2 \]

8. **Finding the Ratio of Momentum**:
Now, we can find the ratio of their momenta:
\[ \frac{P_1}{P_2} = \frac{2 v_1}{6 v_2} = \frac{2}{6} \cdot \frac{v_1}{v_2} = \frac{1}{3} \cdot 3 = 1 \]

9. **Final Answer**:
Therefore, the ratio of their linear momentum is:
\[ P_1 : P_2 = 1 : 1 \]

### Conclusion:
The ratio of the linear momentum of the two balls is \( 1 : 1 \).