Two particles having mass `M_1 =4gm ,M_2 =16gm`. If kinetic energy of both particle are equal then ratio of their momentum is `n :2` then `n` is

Options

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

Answer

To solve the problem, we need to find the ratio of the momentum of two particles given that their kinetic energies are equal.

### Step-by-Step Solution:

1. **Identify the Given Data**:
- Mass of particle 1, \( M_1 = 4 \, \text{g} \)
- Mass of particle 2, \( M_2 = 16 \, \text{g} \)
- Kinetic energy of both particles is equal.

2. **Write the Formula for Kinetic Energy**:
The kinetic energy (KE) of a particle is given by:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass and \( v \) is the velocity of the particle.

3. **Express Velocity in Terms of Kinetic Energy**:
Rearranging the kinetic energy formula to find velocity:
\[ v = \sqrt{\frac{2 \cdot KE}{m}} \]

4. **Write the Momentum Formula**:
The momentum \( p \) of a particle is given by:
\[ p = m v \]

5. **Substitute Velocity into the Momentum Formula**:
For particle 1:
\[ p_1 = M_1 \cdot v_1 = M_1 \cdot \sqrt{\frac{2 \cdot KE}{M_1}} = \sqrt{2 \cdot M_1 \cdot KE} \]
For particle 2:
\[ p_2 = M_2 \cdot v_2 = M_2 \cdot \sqrt{\frac{2 \cdot KE}{M_2}} = \sqrt{2 \cdot M_2 \cdot KE} \]

6. **Find the Ratio of Momenta**:
The ratio of the momenta \( \frac{p_1}{p_2} \) is:
\[ \frac{p_1}{p_2} = \frac{\sqrt{2 \cdot M_1 \cdot KE}}{\sqrt{2 \cdot M_2 \cdot KE}} = \frac{\sqrt{M_1}}{\sqrt{M_2}} \]
Since \( KE \) cancels out and \( \sqrt{2} \) also cancels out.

7. **Substitute the Mass Values**:
\[ \frac{p_1}{p_2} = \frac{\sqrt{4}}{\sqrt{16}} = \frac{2}{4} = \frac{1}{2} \]

8. **Express the Ratio in Terms of n**:
We are given that the ratio of their momentum is \( n:2 \). From our calculation:
\[ \frac{p_1}{p_2} = \frac{1}{2} \implies n = 1 \]

### Final Answer:
Thus, the value of \( n \) is \( 1 \).