Two bodies A & B have velocities in the ratio 2 : 1. Mass of the body A is half of the mass of the body B. The kinetic energies are in the ratio of

Options

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

Answer

To solve the problem of finding the ratio of kinetic energies of two bodies A and B, we will follow these steps:

### Step 1: Define the Variables
Let:
- \( m_A \) = mass of body A
- \( m_B \) = mass of body B
- \( v_A \) = velocity of body A
- \( v_B \) = velocity of body B

### Step 2: Establish the Given Ratios
From the problem statement:
- The velocities are in the ratio \( v_A : v_B = 2 : 1 \).
- The mass of body A is half of the mass of body B, so we can express this as:
\[ m_A = \frac{1}{2} m_B \]

### Step 3: Write the Kinetic Energy Formulas
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
Thus, the kinetic energies for bodies A and B can be expressed as:
- For body A:
\[ KE_A = \frac{1}{2} m_A v_A^2 \]
- For body B:
\[ KE_B = \frac{1}{2} m_B v_B^2 \]

### Step 4: Substitute the Mass and Velocity Ratios
Substituting \( m_A \) and the velocity ratio into the kinetic energy equations:
- Substitute \( m_A = \frac{1}{2} m_B \) into \( KE_A \):
\[ KE_A = \frac{1}{2} \left(\frac{1}{2} m_B\right) v_A^2 = \frac{1}{4} m_B v_A^2 \]
- Since \( v_A : v_B = 2 : 1 \), we can express \( v_A \) in terms of \( v_B \):
\[ v_A = 2 v_B \]
- Substitute \( v_A \) into the kinetic energy of A:
\[ KE_A = \frac{1}{4} m_B (2 v_B)^2 = \frac{1}{4} m_B \cdot 4 v_B^2 = m_B v_B^2 \]

### Step 5: Write the Ratio of Kinetic Energies
Now, we can write the ratio of the kinetic energies:
\[ \frac{KE_A}{KE_B} = \frac{m_B v_B^2}{\frac{1}{2} m_B v_B^2} \]
The \( m_B \) and \( v_B^2 \) terms cancel out:
\[ \frac{KE_A}{KE_B} = \frac{1}{\frac{1}{2}} = 2 \]

### Step 6: Final Ratio of Kinetic Energies
Thus, the ratio of the kinetic energies of A and B is:
\[ KE_A : KE_B = 2 : 1 \]

### Conclusion
The kinetic energies of bodies A and B are in the ratio of \( 2 : 1 \).

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