A bullet is fired through a board, 14.0 cm thick, with its line of motion perpendicular to the face of the board. If it enters with a speed of 450 m/s and emerges with a speed of 220 m/s, what is the bullet's acceleration as it passes through the board ?

Options

A. `-500 km//s^(2)`
B. `-550 km//s^(2)`
C. `-360 km//s^(2)`
D. `-520 km//s^(2)`

Answer

To solve the problem of finding the bullet's acceleration as it passes through the board, we can use the third equation of motion. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Given Values
- Thickness of the board (distance, \( s \)): 14.0 cm = 0.14 m (convert to meters)
- Initial velocity of the bullet (\( u \)): 450 m/s (speed at entry)
- Final velocity of the bullet (\( v \)): 220 m/s (speed at exit)

### Step 2: Use the Third Equation of Motion
The third equation of motion relates the initial velocity, final velocity, acceleration, and displacement:
\[ v^2 = u^2 + 2as \]
Where:
- \( v \) = final velocity
- \( u \) = initial velocity
- \( a \) = acceleration
- \( s \) = displacement

### Step 3: Rearrange the Equation to Solve for Acceleration
We can rearrange the equation to solve for acceleration \( a \):
\[ a = \frac{v^2 - u^2}{2s} \]

### Step 4: Substitute the Known Values
Now, substitute the known values into the equation:
\[ a = \frac{(220)^2 - (450)^2}{2 \times 0.14} \]

### Step 5: Calculate the Values
First, calculate \( (220)^2 \) and \( (450)^2 \):
- \( (220)^2 = 48400 \)
- \( (450)^2 = 202500 \)

Now substitute these values into the equation:
\[ a = \frac{48400 - 202500}{2 \times 0.14} \]
\[ a = \frac{-154100}{0.28} \]
\[ a = -550357.14 \, \text{m/s}^2 \]

### Step 6: Convert to Kilometers per Second Squared
To convert the acceleration from m/s² to km/s², we divide by 1000:
\[ a = -550.36 \, \text{km/s}^2 \]

### Step 7: Interpret the Result
The negative sign indicates that the bullet is decelerating (retardation) as it passes through the board.

### Final Answer
The bullet's acceleration as it passes through the board is approximately:
\[ a \approx -550.36 \, \text{km/s}^2 \]

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