Two masses of 4kg and 16kg are moving with equal K.E. The ratio of magnitude of the linear momentum is

Options

A. `1:2`
B. `1:4`
C. `1:sqrt2`
D. `sqrt2:1`

Explanation

To find the ratio of the magnitudes of linear momentum for two masses (4 kg and 16 kg) moving with equal kinetic energy, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between kinetic energy and momentum**:
The kinetic energy (K.E.) of an object is given by the formula:
\[ K.E. = \frac{1}{2} mv^2 \]
The momentum (P) of an object is given by:
\[ P = mv \]
We can express velocity (v) in terms of momentum:
\[ v = \frac{P}{m} \]

2. **Express kinetic energy in terms of momentum**:
Substitute \(v\) in the kinetic energy formula:
\[ K.E. = \frac{1}{2} m \left(\frac{P}{m}\right)^2 = \frac{P^2}{2m} \]
Thus, we can express kinetic energy as:
\[ K.E. = \frac{P^2}{2m} \]

3. **Set up the equations for both masses**:
For mass \(m_1 = 4 \, \text{kg}\) and mass \(m_2 = 16 \, \text{kg}\), since they have equal kinetic energy (\(K_1 = K_2\)), we can write:
\[ K_1 = \frac{P_1^2}{2m_1} \quad \text{and} \quad K_2 = \frac{P_2^2}{2m_2} \]
Setting \(K_1 = K_2\):
\[ \frac{P_1^2}{2 \cdot 4} = \frac{P_2^2}{2 \cdot 16} \]

4. **Simplify the equation**:
This simplifies to:
\[ \frac{P_1^2}{8} = \frac{P_2^2}{32} \]
Cross-multiplying gives:
\[ 32P_1^2 = 8P_2^2 \]
Dividing both sides by 8:
\[ 4P_1^2 = P_2^2 \]

5. **Find the ratio of momenta**:
Taking the square root of both sides:
\[ \frac{P_1}{P_2} = \frac{1}{2} \]
Therefore, the ratio of the magnitudes of linear momentum is:
\[ P_1 : P_2 = 1 : 2 \]

### Final Answer:
The ratio of the magnitudes of the linear momentum is \( \frac{1}{2} \) or \( 1:2 \).