A bullet from a gun is fired on a rectangular wooden block with velocity $u$. When bullet travels $24$ cm through the block along its length horizontally, velocity of bullet becomes $\dfrac{u}{3}$. Then it further penetrates into the block in the same direction before coming to rest exactly at the other end of the block. The total length of the block is :
Answer: (A) $27$ cm
Concept. Uniform retardation: $v^2 = u^2 - 2as$. Compare the first $24$ cm with the whole length.
Step 1. First $24$ cm.
$$\left(\frac{u}{3}\right)^2 = u^2 - 2a(24) \;\Rightarrow\; 48a = \frac{8u^2}{9}$$
Step 2. Whole length. From $u$ to rest: $u^2 = 2aL$.
Step 3. Divide.
$$\frac{2aL}{48a} = \frac{u^2}{8u^2/9} = \frac98 \;\Rightarrow\; L = 24 \times \frac98 = 27\ \text{cm}$$
Answer. $27$ cm (Option A).
Solution by Sreeraj P, M.Sc Physics