A target is made of two plates, one of wood and the other of iron. The thickness of the wooden plate is 4 cm and that of iron plate is 2 cm. A bullet fired goes through the wood first and then penetrates 1 cm into iron. A similar bullet fired with the same velocity from opposite direction goes through iron first and then penetrates 2 cm into wood. If `a_(1)` and `a_(2)` be the retardations offered to the bullet by wood and iron plates, respectively, then

Options

A. `a_(1)=2a_(2)`
B. `a_(2)=2a_(1)`
C. `a_(1)=a_(2)`
D. Data insufficient

Answer

To solve the problem, we need to analyze the motion of the bullet as it penetrates through the wooden and iron plates. We will derive equations based on the kinematic equations of motion and then find the relationship between the retardations offered by the wooden plate (A1) and the iron plate (A2).

### Step-by-Step Solution:

1. **Understanding the Problem:**
- The wooden plate has a thickness of 4 cm.
- The iron plate has a thickness of 2 cm.
- A bullet first penetrates the wood and then goes 1 cm into the iron.
- A similar bullet fired from the opposite direction penetrates the iron first and then goes 2 cm into the wood.

2. **Define Variables:**
- Let the initial velocity of the bullet be \( V \).
- Let \( V_1 \) be the velocity of the bullet after penetrating the wooden plate.
- Let \( V_2 \) be the velocity of the bullet after penetrating the iron plate.

3. **First Case (Wood first, then Iron):**
- For the wooden plate:
- Using the equation of motion:
\[ V_1^2 = V^2 - 2A_1 \cdot 4 \quad \text{(1)} \]
- For the iron plate (1 cm penetration):
- The bullet stops after penetrating 1 cm:
\[ 0 = V_1^2 - 2A_2 \cdot 1 \quad \text{(2)} \]

4. **Second Case (Iron first, then Wood):**
- For the iron plate (2 cm penetration):
- Using the equation of motion:
\[ V_2^2 = V^2 - 2A_2 \cdot 2 \quad \text{(3)} \]
- For the wooden plate (2 cm penetration):
- The bullet stops after penetrating 2 cm:
\[ 0 = V_2^2 - 2A_1 \cdot 2 \quad \text{(4)} \]

5. **Eliminating Variables:**
- From equations (1) and (2):
- Substitute \( V_1^2 \) from (1) into (2):
\[ 0 = (V^2 - 8A_1) - 2A_2 \quad \Rightarrow \quad 8A_1 + 2A_2 = V^2 \quad \text{(5)} \]
- From equations (3) and (4):
- Substitute \( V_2^2 \) from (3) into (4):
\[ 0 = (V^2 - 4A_2) - 4A_1 \quad \Rightarrow \quad 4A_1 + 4A_2 = V^2 \quad \text{(6)} \]

6. **Setting Equations Equal:**
- Set equations (5) and (6) equal to each other:
\[ 8A_1 + 2A_2 = 4A_1 + 4A_2 \]
- Rearranging gives:
\[ 8A_1 - 4A_1 = 4A_2 - 2A_2 \quad \Rightarrow \quad 4A_1 = 2A_2 \]
- Simplifying gives:
\[ A_2 = 2A_1 \]

7. **Conclusion:**
- The relationship between the retardations is:
\[ A_2 = 2A_1 \]