To solve for the acceleration of the bullet as it passes through the board, we can use the kinematic equation that relates the initial velocity, final velocity, displacement, and acceleration:

$v_f^2 = v_i^2 + 2a s$

Given:

• Initial velocity, $v_i = 400 \, \text{m/s}$
• Final velocity, $v_f = 200 \, \text{m/s}$
• Thickness of the board (displacement), $s = 10.0 \, \text{cm} = 0.10 \, \text{m}$

Finding the Acceleration $a$:

Rearrange the equation to solve for $a$:

$v_f^2 = v_i^2 + 2a s$

$a = \frac{v_f^2 - v_i^2}{2s}$

Now, substitute the known values:

$a = \frac{(200 \, \text{m/s})^2 - (400 \, \text{m/s})^2}{2 \times 0.10 \, \text{m}}$

Calculate the squares of the velocities:

$a = \frac{40000 \, \text{m}^2/\text{s}^2 - 160000 \, \text{m}^2/\text{s}^2}{0.20 \, \text{m}}$

$a = \frac{-120000 \, \text{m}^2/\text{s}^2}{0.20 \, \text{m}}$

$a = -600000 \, \text{m/s}^2$

Final Answer:

The acceleration of the bullet as it passes through the board is $-600000 \, \text{m/s}^2$.

The negative sign indicates that the bullet is decelerating as it passes through the board.

Would you like more details or have any questions?

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Here are some related questions:

1. What would the acceleration be if the board were twice as thick?
2. How much time does the bullet spend passing through the board?
3. How would the bullet's final velocity change if the initial velocity were 500 m/s?
4. What would be the acceleration if the bullet left the board at 100 m/s instead of 200 m/s?
5. How does the thickness of the board affect the bullet's acceleration?

Tip: When solving physics problems, always check the direction of the acceleration to determine if it's a deceleration (negative) or an acceleration (positive).