Two particles have masses m and 4m and their kinetic energies are in the ratio 2 : 1. What is the ratio of their linear momenta?

Options

A. `1//sqrt2`
B. `1//2`
C. `1//4`
D. `1//16`

Explanation

To solve the problem, we need to find the ratio of the linear momenta of two particles with masses \( m \) and \( 4m \) given that their kinetic energies are in the ratio \( 2:1 \).

### Step-by-Step Solution:

1. **Define the Kinetic Energy of Each Particle**:
- Let the kinetic energy of the first particle (mass \( m \)) be \( K_1 \).
- Let the kinetic energy of the second particle (mass \( 4m \)) be \( K_2 \).
- According to the problem, we have:
\[ \frac{K_1}{K_2} = \frac{2}{1} \]
- This implies \( K_1 = 2K_2 \).

2. **Use the Formula for Kinetic Energy**:
- The kinetic energy \( K \) of an object is given by the formula:
\[ K = \frac{p^2}{2m} \]
- Where \( p \) is the linear momentum and \( m \) is the mass of the object.

3. **Express Kinetic Energies in Terms of Momentum**:
- For the first particle:
\[ K_1 = \frac{p_1^2}{2m} \]
- For the second particle:
\[ K_2 = \frac{p_2^2}{2(4m)} = \frac{p_2^2}{8m} \]

4. **Set Up the Ratio of Kinetic Energies**:
- Using the relationship \( K_1 = 2K_2 \):
\[ \frac{p_1^2}{2m} = 2 \left( \frac{p_2^2}{8m} \right) \]

5. **Simplify the Equation**:
- This simplifies to:
\[ \frac{p_1^2}{2m} = \frac{p_2^2}{4m} \]
- By multiplying both sides by \( 4m \):
\[ 2p_1^2 = p_2^2 \]

6. **Find the Ratio of Linear Momenta**:
- Rearranging gives:
\[ \frac{p_1^2}{p_2^2} = \frac{1}{2} \]
- Taking the square root of both sides:
\[ \frac{p_1}{p_2} = \frac{1}{\sqrt{2}} \]

7. **Express the Ratio**:
- Therefore, the ratio of the linear momenta \( p_1 : p_2 \) is:
\[ p_1 : p_2 = 1 : \sqrt{2} \]

### Final Answer:
The ratio of their linear momenta is \( 1 : \sqrt{2} \).