If a body looses half of its velocity on penetrating 3 cm in a wooden block, then how much will it penetrate more before coming to rest?

Options

A. 1 cm
B. 2 cm
C. 3 cm
D. 4 cm

Answer

To solve the problem, we need to analyze the motion of the body as it penetrates the wooden block and loses its velocity. We can break down the solution into the following steps:

### Step 1: Understand the Initial Conditions
The body initially has a velocity \( v_0 \) and penetrates 3 cm into the wooden block, after which its velocity is reduced to \( \frac{v_0}{2} \).

### Step 2: Use the Third Equation of Motion
We will apply the third equation of motion, which is given by:
\[ v^2 = u^2 + 2as \]
where:
- \( v \) is the final velocity,
- \( u \) is the initial velocity,
- \( a \) is the acceleration,
- \( s \) is the displacement.

For the first part of the penetration:
- \( u = v_0 \)
- \( v = \frac{v_0}{2} \)
- \( s = 3 \, \text{cm} = 0.03 \, \text{m} \)

Substituting into the equation:
\[ \left(\frac{v_0}{2}\right)^2 = v_0^2 + 2(-a)(0.03) \]
This simplifies to:
\[ \frac{v_0^2}{4} = v_0^2 - 0.06a \]

### Step 3: Rearranging the Equation
Rearranging gives us:
\[ 0.06a = v_0^2 - \frac{v_0^2}{4} \]
\[ 0.06a = \frac{3v_0^2}{4} \]
Now, solving for \( a \):
\[ a = \frac{3v_0^2}{4 \times 0.06} = \frac{3v_0^2}{0.24} = 12.5v_0^2 \]

### Step 4: Calculate the Further Penetration
Now we need to find how much further the body will penetrate before coming to rest. The initial velocity for this part is \( \frac{v_0}{2} \) and the final velocity is 0. Using the same equation:
\[ 0^2 = \left(\frac{v_0}{2}\right)^2 + 2(-a)x \]
Substituting \( a = 12.5v_0^2 \):
\[ 0 = \frac{v_0^2}{4} - 25v_0^2x \]
This simplifies to:
\[ 25v_0^2x = \frac{v_0^2}{4} \]
Dividing both sides by \( v_0^2 \) (assuming \( v_0 \neq 0 \)):
\[ 25x = \frac{1}{4} \]
Thus:
\[ x = \frac{1}{100} = 0.01 \, \text{m} = 1 \, \text{cm} \]

### Step 5: Conclusion
The body will penetrate an additional 1 cm before coming to rest.