A bullet fired into a wooden block loses half of its velocity after penetrating 60cm. It comes to rest after penetrating a further distance of

Options

A. 22cm
B. 20cm
C. 24cm
D. 26cm

Answer

To solve the problem step by step, we will analyze the motion of the bullet as it penetrates the wooden block.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- Let the initial velocity of the bullet be \( u \).
- After penetrating 60 cm, the bullet's velocity is reduced to \( \frac{u}{2} \).

2. **Identifying the Motion**:
- The bullet penetrates the block from point B to point C (60 cm) and then continues to point D until it comes to rest.
- We will denote the retardation (deceleration) experienced by the bullet as \( a \).

3. **Applying the First Equation of Motion**:
- For the first part of the motion (from B to C):
\[ v^2 - u^2 = 2as \]
- Here, \( v = \frac{u}{2} \), \( u = u \), \( s = 60 \, \text{cm} \), and \( a \) is negative because it is retardation.
- Substituting these values into the equation:
\[ \left(\frac{u}{2}\right)^2 - u^2 = -2a(60) \]
- Simplifying this gives:
\[ \frac{u^2}{4} - u^2 = -120a \]
- This simplifies to:
\[ -\frac{3u^2}{4} = -120a \]
- Therefore:
\[ \frac{3u^2}{4} = 120a \quad \Rightarrow \quad u^2 = \frac{160a}{3} \]

4. **Analyzing the Second Part of the Motion (C to D)**:
- Now, we analyze the motion from point C to point D where the bullet comes to rest.
- The initial velocity at point C is \( \frac{u}{2} \) and final velocity \( v = 0 \).
- Let the distance from C to D be \( x \).
- Using the same equation of motion:
\[ v^2 - u^2 = 2as \]
- Substituting the known values:
\[ 0 - \left(\frac{u}{2}\right)^2 = -2a(x) \]
- This simplifies to:
\[ -\frac{u^2}{4} = -2ax \]
- Therefore:
\[ \frac{u^2}{4} = 2ax \quad \Rightarrow \quad x = \frac{u^2}{8a} \]

5. **Substituting for \( u^2 \)**:
- Now we substitute \( u^2 \) from the earlier equation:
\[ x = \frac{\frac{160a}{3}}{8a} \]
- Simplifying this gives:
\[ x = \frac{160}{24} = \frac{20}{3} \, \text{cm} \]

6. **Final Calculation**:
- Thus, the bullet will penetrate an additional distance of \( 20 \, \text{cm} \) before coming to rest.

### Final Answer:
The bullet will come to rest after penetrating an additional distance of **20 cm**.