A bullet is fired normally towards an immovable wooden block. If it loses `25 %` of its kinetic energy in penetrating through the block at thickness `x`, the distance penetrated by the bullet into the block is

Options

A. 4x
B. 6x
C. 8x
D. 2x

Answer

To solve the problem, we need to analyze the situation using the concepts of kinetic energy and work done.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A bullet is fired towards an immovable wooden block and penetrates a distance \( x \).
- During this penetration, the bullet loses \( 25\% \) of its kinetic energy.

2. **Kinetic Energy Loss**:
- Let the initial kinetic energy of the bullet be \( KE = \frac{1}{2} mv^2 \), where \( m \) is the mass of the bullet and \( v \) is its velocity.
- The bullet loses \( 25\% \) of its kinetic energy while penetrating the block. Therefore, the energy lost is:
\[ \text{Energy lost} = 0.25 \times KE = 0.25 \times \frac{1}{2} mv^2 = \frac{1}{8} mv^2 \]

3. **Work Done by the Bullet**:
- According to the work-energy theorem, the work done by the bullet in penetrating the block is equal to the energy lost.
- The work done can also be expressed as:
\[ W = F \cdot x \]
- Where \( F \) is the average resistive force exerted by the block on the bullet.

4. **Equating Work Done to Energy Lost**:
- Setting the work done equal to the energy lost gives us:
\[ F \cdot x = \frac{1}{8} mv^2 \]

5. **Total Kinetic Energy Consumption**:
- When the bullet penetrates the block completely (let's say it penetrates a distance \( S \)), all of its kinetic energy will be converted into work done against the resistive force:
\[ F \cdot S = KE = \frac{1}{2} mv^2 \]

6. **Relating the Two Equations**:
- From the two equations we have:
- \( F \cdot x = \frac{1}{8} mv^2 \) (1)
- \( F \cdot S = \frac{1}{2} mv^2 \) (2)

7. **Dividing Equation (2) by Equation (1)**:
- Dividing the second equation by the first gives:
\[ \frac{F \cdot S}{F \cdot x} = \frac{\frac{1}{2} mv^2}{\frac{1}{8} mv^2} \]
- Simplifying this, we find:
\[ \frac{S}{x} = \frac{\frac{1}{2}}{\frac{1}{8}} = 4 \]
- Therefore, we have:
\[ S = 4x \]

### Final Answer:
The distance penetrated by the bullet into the block is \( S = 4x \).