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 further distance penetrated by the bullet into the block is

Options

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

Explanation

To solve the problem step by step, we will use the work-energy theorem, which relates the work done on an object to its change in kinetic energy.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- A bullet is fired towards a wooden block.
- The bullet loses 25% of its kinetic energy after penetrating a distance \( x \).

2. **Kinetic Energy Loss**:
- Let the initial kinetic energy of the bullet be \( KE_i \).
- The bullet loses 25% of its kinetic energy while penetrating through the block, so the energy lost is:
\[ \Delta KE = \frac{1}{4} KE_i \]
- Therefore, the kinetic energy after penetrating distance \( x \) is:
\[ KE_f = KE_i - \Delta KE = KE_i - \frac{1}{4} KE_i = \frac{3}{4} KE_i \]

3. **Applying the Work-Energy Theorem**:
- According to the work-energy theorem:
\[ W = \Delta KE \]
- The work done by the opposing force (due to the block) while penetrating distance \( x \) is:
\[ W = F \cdot x \]
- Here, \( F \) is the average resistive force exerted by the block.

4. **Setting Up the Equation**:
- Since the work done is equal to the change in kinetic energy:
\[ F \cdot x = \frac{1}{4} KE_i \]

5. **Finding Total Distance Penetrated**:
- Now, let \( S \) be the total distance the bullet penetrates into the block until it comes to rest.
- The total change in kinetic energy when the bullet comes to rest is:
\[ \Delta KE_{total} = KE_i \]
- The work done over the total distance \( S \) is:
\[ W_{total} = F \cdot S \]
- Setting this equal to the total change in kinetic energy:
\[ F \cdot S = KE_i \]

6. **Relating Work Done Over Distances**:
- From the two equations we have:
\[ F \cdot x = \frac{1}{4} KE_i \]
\[ F \cdot S = KE_i \]
- Dividing the second equation by the first gives:
\[ \frac{F \cdot S}{F \cdot x} = \frac{KE_i}{\frac{1}{4} KE_i} \]
- This simplifies to:
\[ \frac{S}{x} = 4 \]
- Therefore, we find:
\[ S = 4x \]

7. **Conclusion**:
- The total distance penetrated by the bullet into the block is \( 4x \).

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