A body of mass a moving with a velocity b strikes a body of mass c and gets embedded 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 step by step, we will apply the principle of conservation of momentum.

### Step 1: Identify the masses and their velocities
- Let the mass of the first body (A) be \( m_A = A \).
- The velocity of the first body (A) is \( u_A = B \).
- Let the mass of the second body (C) be \( m_C = C \).
- The initial velocity of the second body (C) is \( u_C = 0 \) (since it is at rest).

### Step 2: Write the expression for initial momentum
The total initial momentum \( P_i \) of the system before the collision is given by:
\[ P_i = m_A \cdot u_A + m_C \cdot u_C \]
Substituting the values:
\[ P_i = A \cdot B + C \cdot 0 = A \cdot B \]

### Step 3: Write the expression for final momentum
After the collision, the two bodies get embedded into each other, and we can treat them as a single body with a combined mass:
\[ m_{total} = m_A + m_C = A + C \]
Let \( V \) be the common velocity of the combined mass after the collision. The final momentum \( P_f \) is then:
\[ P_f = m_{total} \cdot V = (A + C) \cdot V \]

### Step 4: Apply the conservation of momentum
According to the conservation of momentum:
\[ P_i = P_f \]
Substituting the expressions we derived:
\[ A \cdot B = (A + C) \cdot V \]

### Step 5: Solve for the final velocity \( V \)
Rearranging the equation to solve for \( V \):
\[ V = \frac{A \cdot B}{A + C} \]

### Conclusion
Thus, the velocity of the system after the collision is:
\[ V = \frac{A \cdot B}{A + C} \]

This matches with option 2 from the question.

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