Question

A bullet of mass a and velocity b is fired into a large block of wood of mass c, the final velocity of the system is
(A). $\dfrac{b}{a+c}a$
(B). $\dfrac{a+b}{c}a$
(C). $\dfrac{a}{a+c}a$
(D). $\dfrac{a+c}{a}b$

Answer 1

Hint: Observe that the system, which includes the bullet and the box, does not experience any external force on it. Therefore, apply conservation of momentum for this system to find the final velocity of the two after collision.

Formula used: Formula for conservation of momentum:
${{P}_{initial}}={{P}_{final}}$
Formula for momentum:
$\text{momentum }=\text{mass }\times \text{velocity}$

Complete step by step answer:
Let us understand what is happening in the question A bullet of mass $a$ is reaching towards a box $c$ with a constant velocity $b$ . Due to collision between the bullet and the box, the bullet will get stuck in the box. Finally, both, the bullet and the box, move with some velocity. Refer to the figure.

Considering the system which consists of the bullet and the box, observe that there is no external force which acts on the system. External force is defined as any disturbance such as an impulse caused by something that is not a part of the system.
When a system does not experience an external force, it conserves the total momentum. Momentum refers to the quantity of motion that an object has. The formula of the momentum is
$\text{momentum }=\text{mass }\times \text{velocity}$
Conservation of momentum means the total momentum of the system before collision is the same as the total momentum afterwards. Initially, the velocity of the bullet was b and the box was at rest (zero velocity). Therefore, the total momentum before the collision was
$\begin{align}
  & {{\text{P}}_{i}}\text{= mass of bullet}\times \text{velocity of the bullet+mass of box}\times \text{velocity of the box} \\
 & {{\text{P}}_{i}}=ab+0c \\
 & {{P}_{i}}=ab \\
\end{align}$Calculating momentum after collision. Since bullets and box are stuck together and travel with the same velocity $v$. The final momentum of the system is
$\begin{align}
  & {{\text{P}}_{f}}\text{= combined mass}\times \text{combined velocity} \\
 & {{\text{P}}_{f}}=(a+c)v \\
\end{align}$According to the law of conservation of momentum
$\begin{align}
  & {{P}_{initial}}={{P}_{final}} \\
 & ab=(a+c)v \\
 & v=\dfrac{b}{(a+c)}a \\
\end{align}$Therefore, the answer to this question is A. $\dfrac{b}{a+c}a$.

Note: Another way to solve this problem is to eliminate the wrong options looking at the dimensions. Since velocity is asked, the correct option must have dimensions of velocity. In option B. $\dfrac{a+b}{c}a$, mass $a$ and velocity $b$ are added which is dimensionally incorrect. Option C. $\dfrac{a}{a+c}a$ has dimension of mass and thus, is incorrect.