. A bullet moving with a velocity of 200 cm / s penetrates a wooden block and comes to rest after traversing 4 cm inside it. What velocity is needed for travelling distance of 9cm in same block

Options

A. 100 m/s
B. 136.2 cm/s
C. 300 cm/s
D. 250 cm/s

Answer

To solve the problem step by step, we will use the equations of motion.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Initial velocity of the bullet, \( u = 200 \, \text{cm/s} \)
- Final velocity of the bullet after penetrating 4 cm, \( v = 0 \, \text{cm/s} \)
- Distance penetrated, \( s_1 = 4 \, \text{cm} \)
- Distance to find the velocity for, \( s_2 = 9 \, \text{cm} \)

2. **Use the First Equation of Motion:**
We will use the equation:
\[ v^2 = u^2 + 2as \]
Rearranging it for acceleration \( a \):
\[ a = \frac{v^2 - u^2}{2s} \]

3. **Calculate the Acceleration (Retardation) when the bullet penetrates 4 cm:**
Substituting \( u = 200 \, \text{cm/s} \), \( v = 0 \, \text{cm/s} \), and \( s_1 = 4 \, \text{cm} \):
\[ a = \frac{0^2 - (200)^2}{2 \times 4} = \frac{-40000}{8} = -5000 \, \text{cm/s}^2 \]
(The negative sign indicates retardation.)

4. **Use the Acceleration to Find the Required Velocity for 9 cm:**
Now, we need to find the final velocity \( v \) when the bullet travels a distance of \( s_2 = 9 \, \text{cm} \). We will use the same equation:
\[ v^2 = u^2 + 2as \]
Here, the initial velocity \( u \) is the velocity we need to find, and \( a = -5000 \, \text{cm/s}^2 \) (retardation), and \( s = 9 \, \text{cm} \).

5. **Substituting the Values:**
Since the bullet starts from rest in the block, we can assume \( u \) is the velocity we are trying to find:
\[ v^2 = u^2 + 2(-5000)(9) \]
\[ v^2 = u^2 - 90000 \]

6. **Setting the Equation for Final Velocity:**
We need to find the value of \( u \) such that it comes to rest after traveling 9 cm. Thus, we set \( v = 0 \):
\[ 0 = u^2 - 90000 \]
\[ u^2 = 90000 \]
\[ u = \sqrt{90000} = 300 \, \text{cm/s} \]

### Final Answer:
The velocity needed for the bullet to travel a distance of 9 cm in the same block is **300 cm/s**.