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![### Understanding the Motion of a Bullet and Block System
**Problem Description:**
Consider a scenario where a bullet impacts and embeds itself into a block. Assume you know the speed of the bullet just before the impact and the mass of the bullet.
The task is to determine the angle \(\theta\) using both words and equations.
**Diagram Explanation:**
The diagram illustrates a system where a bullet has impacted a block, which is part of a pendulum set-up. The block (including the bullet) swings upwards after the collision.
- The pendulum is initially vertical.
- After the collision, the pendulum swings to a new position, forming an angle \(\theta\) with the vertical.
- **L** is the length of the pendulum.
- **h** is the vertical height the block rises after the collision.
- **L - h** depicts the remaining vertical component after the block has swung upward.
- The bottom of the diagram shows the center of the pendulum at the highest point of swing.
**Determining the Angle**
1. **Using Energy Conservation:**
- When the bullet embeds into the block, the kinetic energy of the system is partially converted into potential energy at the peak of the swing.
- Formula for potential energy at the highest point:
\[
mgh
\]
- Where \(m\) is the mass of the bullet-block combination, \(g\) is the acceleration due to gravity, and \(h\) is the height.
2. **Using Momentum Conservation:**
- Momentum is conserved during the bullet-block collision.
- Initial momentum:
\[
m_{\text{bullet}} \cdot v_{\text{bullet}}
\]
Where \(v_{\text{bullet}}\) is the speed of the bullet before impact.
- Final momentum (just after impact):
\[
(m_{\text{bullet}} + m_{\text{block}}) \cdot v
\]
Solving for \(v\) gives the velocity just after impact.
3. **Calculate \(h\):**
- \(h\) can be calculated using trigonometry:
\[
h = L - L \cdot \cos(\theta)
\]
4. **Equation for \(\theta\):**
- By equating kinetic and potential energies, or using \(h\) from above, solve for \(\](https://content.bartleby.com/qna-images/question/c247148b-ac88-4d23-ab16-d846cf9acaa7/beacc8a0-47dd-4b4c-9a94-a639964a6fa1/d4rx2wc_processed.jpeg)
Transcribed Image Text:### Understanding the Motion of a Bullet and Block System
**Problem Description:**
Consider a scenario where a bullet impacts and embeds itself into a block. Assume you know the speed of the bullet just before the impact and the mass of the bullet.
The task is to determine the angle \(\theta\) using both words and equations.
**Diagram Explanation:**
The diagram illustrates a system where a bullet has impacted a block, which is part of a pendulum set-up. The block (including the bullet) swings upwards after the collision.
- The pendulum is initially vertical.
- After the collision, the pendulum swings to a new position, forming an angle \(\theta\) with the vertical.
- **L** is the length of the pendulum.
- **h** is the vertical height the block rises after the collision.
- **L - h** depicts the remaining vertical component after the block has swung upward.
- The bottom of the diagram shows the center of the pendulum at the highest point of swing.
**Determining the Angle**
1. **Using Energy Conservation:**
- When the bullet embeds into the block, the kinetic energy of the system is partially converted into potential energy at the peak of the swing.
- Formula for potential energy at the highest point:
\[
mgh
\]
- Where \(m\) is the mass of the bullet-block combination, \(g\) is the acceleration due to gravity, and \(h\) is the height.
2. **Using Momentum Conservation:**
- Momentum is conserved during the bullet-block collision.
- Initial momentum:
\[
m_{\text{bullet}} \cdot v_{\text{bullet}}
\]
Where \(v_{\text{bullet}}\) is the speed of the bullet before impact.
- Final momentum (just after impact):
\[
(m_{\text{bullet}} + m_{\text{block}}) \cdot v
\]
Solving for \(v\) gives the velocity just after impact.
3. **Calculate \(h\):**
- \(h\) can be calculated using trigonometry:
\[
h = L - L \cdot \cos(\theta)
\]
4. **Equation for \(\theta\):**
- By equating kinetic and potential energies, or using \(h\) from above, solve for \(\
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