Two bodies of mass `1 kg and 2 kg` have equal momentum. The ratio of their kinetic energies is:

Options

A. `1:1`
B. `2:1`
C. `1:3`
D. `3:1`

Explanation

To solve the problem, we need to find the ratio of the kinetic energies of two bodies with masses \( m_1 = 1 \, \text{kg} \) and \( m_2 = 2 \, \text{kg} \) that have equal momentum.

### Step-by-Step Solution:

1. **Understanding Momentum**:
The momentum \( P \) of an object is given by the formula:
\[ P = m \cdot v \]
where \( m \) is the mass and \( v \) is the velocity.

2. **Setting Up the Equation**:
Since the two bodies have equal momentum, we can write:
\[ P_1 = P_2 \]
This translates to:
\[ m_1 \cdot v_1 = m_2 \cdot v_2 \]
Substituting the masses:
\[ 1 \cdot v_1 = 2 \cdot v_2 \]

3. **Solving for Velocity Ratio**:
Rearranging the equation gives:
\[ v_1 = 2 \cdot v_2 \]
Thus, the ratio of their velocities is:
\[ \frac{v_1}{v_2} = 2 \]

4. **Kinetic Energy Formula**:
The kinetic energy \( KE \) of an object is given by:
\[ KE = \frac{1}{2} m v^2 \]

5. **Calculating Kinetic Energies**:
For the first body:
\[ KE_1 = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} \cdot 1 \cdot v_1^2 = \frac{1}{2} v_1^2 \]
For the second body:
\[ KE_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} \cdot 2 \cdot v_2^2 = v_2^2 \]

6. **Finding the Ratio of Kinetic Energies**:
Now, we can find the ratio of their kinetic energies:
\[ \frac{KE_1}{KE_2} = \frac{\frac{1}{2} v_1^2}{v_2^2} \]
Substituting \( v_1 = 2 v_2 \):
\[ KE_1 = \frac{1}{2} (2 v_2)^2 = \frac{1}{2} \cdot 4 v_2^2 = 2 v_2^2 \]
Therefore:
\[ \frac{KE_1}{KE_2} = \frac{2 v_2^2}{v_2^2} = 2 \]

7. **Final Ratio**:
The ratio of their kinetic energies is:
\[ KE_1 : KE_2 = 2 : 1 \]

### Final Answer:
The ratio of their kinetic energies is \( 2 : 1 \).