A bullet of mass a is fired with velocity b in a large block of mass c. The final velocity of the system will be

The correct option is ab/a+c.

Let us solve the problem step-by-step using the law of conservation of linear momentum.

Step 1: Identify the initial parameters of the system
- Mass of the bullet = $a$
- Initial velocity of the bullet = $b$
- Mass of the block = $c$
- Since the large block is initially at rest, its initial velocity = $0$

Step 2: Calculate the initial momentum of the system
The total initial momentum ($P_{\text{initial}}$) is the sum of the momentum of the bullet and the momentum of the block:
$P_{\text{initial}}=(\text{mass of bullet}\times \text{velocity of bullet})+(\text{mass of block}\times \text{velocity of block})$
$P_{\text{initial}}=(a\times b)+(c\times 0)$
$P_{\text{initial}}=ab$

Step 3: Calculate the final momentum of the system
After the bullet is fired into the block, it embeds itself in the block, and they move together as a single combined system.
- Combined mass of the system = $a+c$
- Let the final velocity of the combined system be $v$.
The total final momentum ($P_{\text{final}}$) is:
$P_{\text{final}}=(a+c)\times v$

Step 4: Apply the Law of Conservation of Momentum
According to the law of conservation of linear momentum, in the absence of external forces, the total initial momentum equals the total final momentum:
$P_{\text{initial}}=P_{\text{final}}$
$ab=(a+c)v$

Step 5: Solve for the final velocity ($v$)
Rearranging the equation to find $v$:
$v=\frac{ab}{a+c}$

Thus, the final velocity of the system is $ab/(a+c)$, which corresponds to the option ab/a+c.