Question

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 the 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

• A. 24cm
• B. 28cm
• C. 30cm
• D. 27cm

Answer 1

The Correct Option is D

 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 \(\frac{u}{3}\). Then it further penetrates into the block in the same direction before coming to rest exactly at the other end of the block.

Answer 2

Given:
The initial velocity of the bullet u.
After traveling 24 cm horizontally through the block, the velocity of the bullet becomes \(\frac{u}{3}\).
The bullet then further penetrates the block until it comes to rest exactly at the other end.
Starting with the equations:
1. \(\left(\frac{u}{3}\right)^2 = u^2 - 2ax\)
2. \(2ax = u^2 - \frac{u^2}{9}\)

From equation 1:
\(\left(\frac{u}{3}\right)^2 = u^2 - 2ax\)

\(\frac{u^2}{9} = u^2 - 2ax\)

\(u^2 - \frac{u^2}{9} = 2ax\)

\(\frac{8u^2}{9} = 2ax\)

Now, consider the second set of equations:
3. \(v^2 = u^2 + 2ax\)
4. \(0 = u^2 - 2ax_2\)
5. \(2ax_2 = u^2\)

From equation 5:
\(x_2 = \frac{u^2}{2a}\)

Now we compare \(x\) and \(x_2\):
\(\frac{x}{x_2} = \frac{\frac{8u^2}{18a}}{\frac{u^2}{2a}} = \frac{8}{9}\)

\(\frac{24}{x_2} = \frac{8}{9}\)

\(24 \cdot 9 = 8 \cdot x_2\)

\(216 = 8x_2\)

\(x_2 = \frac{216}{8} = 27 \, \text{cm}\)

So, the correct option is (D): 27cm