A body of mass a moving with a velocity b strikes a body of mass c and gets embered into it. The velocity of the systems after collision is

Options

A. `(a+c)/(ab)`
B. `(ab)/(a+c)`
C. `(a)/(b+c)`
D. `(a)/(a+b)`

Explanation

To solve the problem of finding the velocity of the system after an inelastic collision between two bodies, we can use the principle of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Identify the masses and velocities
Let:
- Mass of the first body (A) = m₁
- Velocity of the first body (A) before collision = v₁
- Mass of the second body (C) = m₂
- Velocity of the second body (C) before collision = v₂ (which is 0, as it is at rest)

### Step 2: Write the expression for initial momentum
The initial momentum (p_initial) of the system before the collision can be calculated as:
\[ p_{\text{initial}} = m_1 \cdot v_1 + m_2 \cdot v_2 \]
Substituting the values:
\[ p_{\text{initial}} = m_1 \cdot v + m_2 \cdot 0 = m_1 \cdot v \]

### Step 3: Write the expression for final momentum
After the collision, the two bodies stick together, so the total mass of the system becomes \( m_1 + m_2 \) and let the final velocity of the combined mass be \( v_f \). The final momentum (p_final) is given by:
\[ p_{\text{final}} = (m_1 + m_2) \cdot v_f \]

### Step 4: Apply the conservation of momentum
According to the conservation of momentum:
\[ p_{\text{initial}} = p_{\text{final}} \]
Thus, we have:
\[ m_1 \cdot v = (m_1 + m_2) \cdot v_f \]

### Step 5: Solve for the final velocity \( v_f \)
Rearranging the equation to solve for \( v_f \):
\[ v_f = \frac{m_1 \cdot v}{m_1 + m_2} \]

### Final Answer
The velocity of the system after the collision is:
\[ v_f = \frac{m_1 \cdot v}{m_1 + m_2} \]

---