A body of mass 2 kg moving with a velocity of 3 m/sec collides head on with a body of mass 1 kg moving in opposite direction with a velocity of 4 m/sec. After collision, two bodies stick together and move with a common velocity which in m/sec is equal to

Options

A. (1//4) m/s
B. (1//3) m/s
C. (2//3) m/s
D. (3//4) m/s

Answer

To solve the problem, we will use the principle of conservation of momentum. Here’s a step-by-step breakdown:

### Step 1: Identify the masses and velocities
- Mass of body 1, \( m_1 = 2 \, \text{kg} \)
- Velocity of body 1, \( v_1 = 3 \, \text{m/s} \) (moving in the positive direction)
- Mass of body 2, \( m_2 = 1 \, \text{kg} \)
- Velocity of body 2, \( v_2 = -4 \, \text{m/s} \) (moving in the negative direction)

### Step 2: Write the equation for conservation of momentum
According to the conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
The initial momentum of the system can be calculated as:
\[ \text{Initial Momentum} = m_1 \cdot v_1 + m_2 \cdot v_2 \]
Substituting the values:
\[ \text{Initial Momentum} = (2 \, \text{kg} \cdot 3 \, \text{m/s}) + (1 \, \text{kg} \cdot -4 \, \text{m/s}) \]
\[ \text{Initial Momentum} = 6 \, \text{kg m/s} - 4 \, \text{kg m/s} = 2 \, \text{kg m/s} \]

### Step 3: Calculate the final momentum
After the collision, the two bodies stick together, so their combined mass is:
\[ m_{\text{total}} = m_1 + m_2 = 2 \, \text{kg} + 1 \, \text{kg} = 3 \, \text{kg} \]
Let \( v_c \) be the common velocity after the collision. The final momentum can be expressed as:
\[ \text{Final Momentum} = m_{\text{total}} \cdot v_c = 3 \, \text{kg} \cdot v_c \]

### Step 4: Set the initial momentum equal to the final momentum
Now, we can set the initial momentum equal to the final momentum:
\[ 2 \, \text{kg m/s} = 3 \, \text{kg} \cdot v_c \]

### Step 5: Solve for the common velocity \( v_c \)
Rearranging the equation to find \( v_c \):
\[ v_c = \frac{2 \, \text{kg m/s}}{3 \, \text{kg}} = \frac{2}{3} \, \text{m/s} \]

### Conclusion
The common velocity after the collision is:
\[ v_c = \frac{2}{3} \, \text{m/s} \]