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. (a) 1 cm
B. (b) 2 cm
C. (c) 3 cm
D. (d) 4 cm

Explanation

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

### Step 1: Understand the initial conditions
Let the initial velocity of the body be \( v \). When the body penetrates 3 cm into the wooden block, it loses half of its velocity. So, after penetrating 3 cm, its velocity becomes \( \frac{v}{2} \).

### Step 2: Calculate the initial and final kinetic energies
- The initial kinetic energy (KE_initial) when the body is at velocity \( v \) is:
\[ KE_{\text{initial}} = \frac{1}{2} mv^2 \]
- The final kinetic energy (KE_final) after penetrating 3 cm (when the velocity is \( \frac{v}{2} \)) is:
\[ KE_{\text{final}} = \frac{1}{2} m\left(\frac{v}{2}\right)^2 = \frac{1}{2} m \cdot \frac{v^2}{4} = \frac{1}{8} mv^2 \]

### Step 3: Calculate the change in kinetic energy
The change in kinetic energy (ΔKE) as the body penetrates 3 cm is:
\[ \Delta KE = KE_{\text{initial}} - KE_{\text{final}} = \frac{1}{2} mv^2 - \frac{1}{8} mv^2 = \frac{4}{8} mv^2 - \frac{1}{8} mv^2 = \frac{3}{8} mv^2 \]

### Step 4: Relate work done to penetration distance
According to the work-energy theorem, the work done by the resistive force (F) while penetrating the block is equal to the change in kinetic energy:
\[ F \cdot 3 = \Delta KE \]
Thus,
\[ F \cdot 3 = \frac{3}{8} mv^2 \]

### Step 5: Calculate the kinetic energy when the body has velocity \( \frac{v}{2} \)
Now, when the body has a velocity of \( \frac{v}{2} \), the kinetic energy is:
\[ KE_{\text{initial}} = \frac{1}{8} mv^2 \]
And the final kinetic energy when it comes to rest is 0:
\[ KE_{\text{final}} = 0 \]
Thus, the change in kinetic energy (ΔKE) for this case is:
\[ \Delta KE = KE_{\text{initial}} - KE_{\text{final}} = \frac{1}{8} mv^2 - 0 = \frac{1}{8} mv^2 \]

### Step 6: Relate work done to the new penetration distance
Using the work-energy theorem again, we have:
\[ F \cdot S = \Delta KE \]
Substituting the values:
\[ F \cdot S = \frac{1}{8} mv^2 \]

### Step 7: Divide the equations to find the new penetration distance
From the first case, we have:
\[ F \cdot 3 = \frac{3}{8} mv^2 \]
Dividing the two equations:
\[ \frac{F \cdot S}{F \cdot 3} = \frac{\frac{1}{8} mv^2}{\frac{3}{8} mv^2} \]
This simplifies to:
\[ \frac{S}{3} = \frac{1}{3} \implies S = 1 \text{ cm} \]

### Step 8: Calculate total penetration
The total penetration distance before coming to rest is:
\[ \text{Total penetration} = 3 \text{ cm} + 1 \text{ cm} = 4 \text{ cm} \]

### Final Answer
The body will penetrate a total of **4 cm** into the wooden block before coming to rest.

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