Question

A bullet is fired into a wooden block. How do I calculate the magnitude of opposing resistance and distance moved by the bullet?

Answer 1

To calculate the magnitude of the opposing resistance and the distance moved by the bullet when it is fired into a wooden block, you can use principles from physics, particularly concepts related to energy, momentum, and kinematics. Here is a general approach to solving the problem:

### Assumptions
1. The bullet comes to a stop after entering the block.
2. The block is stationary and does not move.
3. The bullet loses all its kinetic energy to the resistance force from the block.

### Given Variables
- \( m \): mass of the bullet (in kg)
- \( v_0 \): initial velocity of the bullet (in m/s)
- \( F_r \): average opposing resistance force (in N)
- \( d \): distance moved by the bullet in the block (in m)

### Steps to Calculate Opposing Resistance and Distance

1. **Calculate Initial Kinetic Energy of the Bullet**:
The kinetic energy (\( KE \)) of the bullet before it enters the block can be calculated using the formula:
\[
KE = \frac{1}{2} m v_0^2
\]

2. **Set up the Work-Energy Principle**:
The work done by the opposing resistance force (\( F_r \)) as the bullet moves a distance \( d \) is equal to the initial kinetic energy of the bullet. The work \( W \) done by the resistance force can be written as:
\[
W = F_r \cdot d
\]
Setting \( W \) equal to the initial kinetic energy:
\[
F_r \cdot d = \frac{1}{2} m v_0^2
\]

3. **Calculating the Opposing Resistance Force**:
Rearranging the equation gives:
\[
F_r = \frac{\frac{1}{2} m v_0^2}{d}
\]

4. **Determine Distance Moved by the Bullet**:
To find the distance \( d \) the bullet travels into the block, you may need additional information about the opposing force, such as its average value or a relationship with the bullet's penetration depth (which can depend on material properties of the block).

- If the resistance Force is assumed to be constant, you can use the previously derived equations to find the distance if \( F_r \) is known.
- If the resistance force varies, you will need a functional relationship describing how \( F_r \) changes with the distance \( d \).

### Example Calculation
If:
- The mass of the bullet \( m = 0.01 \) kg (10 grams)
- The initial velocity \( v_0 = 300 \) m/s
- Assume \( d = 0.1 \) m (10 cm) for example.

1. **Calculate the Initial Kinetic Energy**:
\[
KE = \frac{1}{2} \cdot 0.01 \cdot 300^2 = 450 J
\]

2. **Calculate the Opposing Resistance Force**:
\[
F_r = \frac{450}{0.1} = 4500 N
\]

This gives an opposing resistance of \( 4500 \) N if the bullet penetrates \( 0.1 \) m into the block.

### Conclusion
You can determine the opposing resistance force and the distance moved by the bullet by applying the work-energy principle under the assumption of constant resistance. If the resistance varies non-linearly, you may need to perform a more complex analysis using integral calculus or empirical data regarding the material penetration properties.