A bullet moving with a velocity of `30sqrt(2) ms^(-1)` is fired into a fixed target. It penetrated into the target to the extent of s metres. If the same bullet is fired into a target of thickness `(s)/(2)` metres and of the same material with the same velocity, then the bullet comes out of the target with velocity

Options

A. `20 ms^(-1)`
B. `30 ms^(-1)`
C. `20sqrt(2)ms^(-1)`
D. `10sqrt(2)ms^(-1)`

Explanation

To solve the problem step by step, we will use the equations of motion to find the final velocity of the bullet after penetrating a target of thickness \( \frac{s}{2} \).

### Step 1: Understanding the problem
We know that a bullet is fired with an initial velocity \( u = 30\sqrt{2} \, \text{m/s} \) and penetrates a target of thickness \( s \) meters, coming to a stop (final velocity \( v = 0 \)). We need to find the final velocity of the bullet when it penetrates a target of thickness \( \frac{s}{2} \).

### Step 2: Calculate the acceleration (deceleration)
Using the equation of motion:
\[ v^2 = u^2 + 2as \]
For the first case, where the bullet comes to rest:
- \( v = 0 \)
- \( u = 30\sqrt{2} \)
- \( s = s \)

Substituting these values into the equation:
\[ 0 = (30\sqrt{2})^2 + 2a(s) \]
This simplifies to:
\[ 0 = 1800 + 2as \]
Rearranging gives:
\[ 2as = -1800 \quad \Rightarrow \quad a = -\frac{900}{s} \, \text{m/s}^2 \]

### Step 3: Apply the same process for the second case
Now, we analyze the second case where the bullet penetrates a target of thickness \( \frac{s}{2} \):
- Initial velocity \( u = 30\sqrt{2} \)
- Distance \( s = \frac{s}{2} \)
- Acceleration \( a = -\frac{900}{s} \)

Using the same equation of motion:
\[ v^2 = u^2 + 2as \]
Substituting the known values:
\[ v^2 = (30\sqrt{2})^2 + 2\left(-\frac{900}{s}\right)\left(\frac{s}{2}\right) \]
This simplifies to:
\[ v^2 = 1800 - 900 \]
Thus:
\[ v^2 = 900 \]
Taking the square root gives:
\[ v = 30 \, \text{m/s} \]

### Conclusion
The final velocity of the bullet after penetrating the target of thickness \( \frac{s}{2} \) is \( 30 \, \text{m/s} \).

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