A bullet of mass 5 g moving at a speed of 200 m/s strikes a rigidly fixed wooden plank of thickness 0.2m normally and passes through it losing half of its kinetic energy. If it again strikes an identical rigidly fixed wooden plank and passes through it, assuming the same resistance in the two planks the ratio of the thermal energies produced in the two planks is

Options

A. `1:1`
B. `1:2`
C. `2:1`
D. `4:1`

Explanation

To solve the problem step by step, we will analyze the situation involving the bullet and the wooden planks.

### Step 1: Calculate the initial kinetic energy of the bullet.
The initial kinetic energy (KE_initial) of the bullet can be calculated using the formula:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the bullet and \( v \) is its velocity.

Given:
- Mass of the bullet, \( m = 5 \, \text{g} = 0.005 \, \text{kg} \) (conversion from grams to kilograms)
- Velocity of the bullet, \( v = 200 \, \text{m/s} \)

Calculating the initial kinetic energy:
\[ KE_{\text{initial}} = \frac{1}{2} \times 0.005 \, \text{kg} \times (200 \, \text{m/s})^2 \]
\[ KE_{\text{initial}} = \frac{1}{2} \times 0.005 \times 40000 = 100 \, \text{J} \]

### Step 2: Determine the kinetic energy after passing through the first plank.
The problem states that the bullet loses half of its kinetic energy when it passes through the first plank. Therefore, the kinetic energy after passing through the plank (KE_final) is:
\[ KE_{\text{final}} = \frac{1}{2} KE_{\text{initial}} = \frac{1}{2} \times 100 \, \text{J} = 50 \, \text{J} \]

### Step 3: Calculate the thermal energy produced in the first plank.
The thermal energy (TE) produced in the first plank is equal to the kinetic energy lost by the bullet:
\[ TE_1 = KE_{\text{initial}} - KE_{\text{final}} = 100 \, \text{J} - 50 \, \text{J} = 50 \, \text{J} \]

### Step 4: Analyze the second plank.
When the bullet strikes the second identical plank, it has a kinetic energy of 50 J (the same as the final kinetic energy from the first plank). It will again lose half of this kinetic energy when passing through the second plank.

### Step 5: Calculate the thermal energy produced in the second plank.
The kinetic energy lost when passing through the second plank is:
\[ KE_{\text{lost, second}} = \frac{1}{2} KE_{\text{final}} = \frac{1}{2} \times 50 \, \text{J} = 25 \, \text{J} \]
Thus, the thermal energy produced in the second plank is:
\[ TE_2 = KE_{\text{lost, second}} = 25 \, \text{J} \]

### Step 6: Find the ratio of thermal energies produced in the two planks.
The ratio of thermal energies produced in the first and second planks is:
\[ \text{Ratio} = \frac{TE_1}{TE_2} = \frac{50 \, \text{J}}{25 \, \text{J}} = 2 \]

### Final Answer:
The ratio of the thermal energies produced in the two planks is \( 2:1 \).

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