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 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. 28 cm
• C. 30cm
• D. 27 cm

Answer 1

The Correct Option is D

To determine the total length of the block, we analyze the bullet's motion. Initially, the bullet is fired with velocity \( u \) and travels a distance of 24 cm through the block while its velocity decreases to \( \frac{u}{3} \). We use the equation of motion:\( v^2 = u^2 + 2as \) 

where \( v = \frac{u}{3} \), \( u = u \), \( s = 24 \, \text{cm} \), and \( a \) is the acceleration (which is negative as the bullet is slowing down).\( \left( \frac{u}{3} \right)^2 = u^2 + 2a \times 24 \)

\( \frac{u^2}{9} = u^2 + 48a \)

\( 48a = \frac{u^2}{9} - u^2 \)

\( 48a = -\frac{8u^2}{9} \)

\( a = -\frac{u^2}{54} \)

Now, we find the additional distance \( x \) the bullet travels before coming to rest:\( 0 = \left( \frac{u}{3} \right)^2 + 2a \times x \)

\( \frac{u^2}{9} = -\frac{u^2}{27}x \)

\( x = 3 \times 9 = 27 \, \text{cm} \)

Hence, the bullet penetrates an additional 3 cm beyond the initial 24 cm to come to rest at the end of the block. Therefore, the total length of the block is:

24 cm (initial penetration)

+ 3 cm (additional distance)

= 27 cm (total length of block)

Answer 2

The correct option is (C): 30cm

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.