Two particles of masses 4kg and 9kg possess kinetic energies in the ratio `(9)/(4)`. The ratio of their linear momentum will be

Options

A. `1:1`
B. `4:9`
C. `2:3`
D. `16:81`

Explanation

To find the ratio of the linear momentum of two particles with given masses and kinetic energy ratios, we can follow these steps:

### Step 1: Understand the given data
We have two particles with the following properties:
- Mass of the first particle, \( m_1 = 4 \, \text{kg} \)
- Mass of the second particle, \( m_2 = 9 \, \text{kg} \)
- Ratio of their kinetic energies, \( \frac{K_1}{K_2} = \frac{9}{4} \)

### Step 2: Write the formula for kinetic energy
The kinetic energy (K.E.) of a particle is given by the formula:
\[ K = \frac{1}{2} mv^2 \]
where \( m \) is the mass and \( v \) is the velocity of the particle.

### Step 3: Express kinetic energies in terms of momentum
The momentum \( p \) of a particle is given by:
\[ p = mv \]
From this, we can express velocity \( v \) in terms of momentum:
\[ v = \frac{p}{m} \]
Substituting this into the kinetic energy formula, we get:
\[ K = \frac{1}{2} m \left(\frac{p}{m}\right)^2 = \frac{p^2}{2m} \]

### Step 4: Set up the ratio of kinetic energies
Using the expression for kinetic energy in terms of momentum, we can write:
\[ K_1 = \frac{p_1^2}{2m_1} \quad \text{and} \quad K_2 = \frac{p_2^2}{2m_2} \]
Now, substituting these into the ratio of kinetic energies:
\[ \frac{K_1}{K_2} = \frac{\frac{p_1^2}{2m_1}}{\frac{p_2^2}{2m_2}} = \frac{p_1^2 \cdot m_2}{p_2^2 \cdot m_1} \]

### Step 5: Substitute the known values
We know that:
\[ \frac{K_1}{K_2} = \frac{9}{4} \]
Thus, we can write:
\[ \frac{9}{4} = \frac{p_1^2 \cdot m_2}{p_2^2 \cdot m_1} \]
Substituting \( m_1 = 4 \, \text{kg} \) and \( m_2 = 9 \, \text{kg} \):
\[ \frac{9}{4} = \frac{p_1^2 \cdot 9}{p_2^2 \cdot 4} \]

### Step 6: Cross-multiply and simplify
Cross-multiplying gives:
\[ 9 \cdot p_2^2 = 4 \cdot p_1^2 \]
Rearranging this gives:
\[ \frac{p_1^2}{p_2^2} = \frac{9}{4} \]

### Step 7: Take the square root to find the ratio of momenta
Taking the square root of both sides:
\[ \frac{p_1}{p_2} = \frac{3}{2} \]

### Conclusion
Thus, the ratio of their linear momentum is:
\[ \frac{p_1}{p_2} = \frac{3}{2} \]