A bullet from a gun is fired on a rectangular wooden block with velocity u. When the bullet travels 24 cm through the block along its length horizontally, velocity of bullet becomes \(\frac{u}{3}\). 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:

Question

A car starts from rest and accelerates uniformly at 3 m/s\textsuperscript{2. What will be its velocity after 5 seconds?}

Answer 1

To solve this problem, we need to determine the total length of the rectangular wooden block that the bullet penetrates. We are given that:

Let's break it down step-by-step:

Determine the deceleration in the first part of the block:
- Using the equation of motion: \( v^2 = u^2 + 2as \)
- Where:
  - \( v = \frac{u}{3} \) is the final velocity after 24 cm,
  - \( u = u \) is the initial velocity,
  - \( s = 24 \text{ cm} \),
  - \( a \) is the deceleration.
- The equation becomes: \[ \left(\frac{u}{3}\right)^2 = u^2 + 2a \times 24 \] \[ \frac{u^2}{9} = u^2 + 48a \]
- Rearranging gives: \[ 48a = \frac{u^2}{9} - u^2 \] \[ 48a = \frac{u^2 - 9u^2}{9} \] \[ 48a = -\frac{8u^2}{9} \] \[ a = -\frac{8u^2}{432} \] \[ a = -\frac{u^2}{54} \]
Determine the total distance the bullet travels:
- When the bullet comes to rest, \( v = 0 \).
- Using the equation again but for the total distance \( S \): \[ 0 = \left(\frac{u}{3}\right)^2 + 2a(s') \] Where \( s' \) is the additional distance after 24 cm.
- The equation becomes: \[ 0 = \frac{u^2}{9} + 2 \left(-\frac{u^2}{54}\right) \times s' \] \[ 0 = \frac{u^2}{9} - \frac{u^2}{27} \times s' \] \[ \frac{u^2}{27} \times s' = \frac{u^2}{9} \] \[ s' = \frac{u^2}{9} \div \frac{u^2}{27} \] \[ s' = 3 \times 24 \] \[ s' = 9 \text{ cm} \]
Calculate the total length of the block:
- The total length of the block is the initial 24 cm + additional 9 cm.
- Total length = 24 cm + 9 cm = 27 cm.

Therefore, the total length of the block is 27 cm.

The Correct Option is D