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To verify if \( \theta = 19.948^\circ \) is correct, let's analyze the setup:

### Given:
- \( m_1 = 0.27 \, \text{kg} \)
- \( L = 1.45 \, \text{m} \)
- \( m_2 = 0.93 \, \text{kg} \)
- \( V_1 = 4.5 \, \text{m/s} \)

### Problem Context:
A ball of mass \( m_1 \) collides horizontally with a hanging ball of mass \( m_2 \), initially at rest. After the collision, \( m_2 \) swings up to reach a maximum height, where it forms an angle \( \theta \) with the vertical.

### Part (b) - Finding \( \theta \):
The angle \( \theta \) can be determined using the principle of conservation of energy. At the highest point, all kinetic energy is converted to potential energy.

1. **Determine the initial kinetic energy transferred to \( m_2 \):**
   Since \( m_1 \) transfers its horizontal momentum to \( m_2 \), let's assume that \( m_2 \) acquires a horizontal velocity \( V_{2} \) after the collision.
   
   Using conservation of momentum:
   \[
   m_1 V_1 = m_2 V_2
   \]
   \[
   V_2 = \frac{m_1 V_1}{m_2} = \frac{0.27 \times 4.5}{0.93} \approx 1.305 \, \text{m/s}
   \]

2. **Convert kinetic energy to potential energy to find the height \( h \):**
   The kinetic energy of \( m_2 \) right after the collision is:
   \[
   KE = \frac{1}{2} m_2 V_2^2
   \]
   Setting \( KE \) equal to the potential energy at the maximum height \( h \):
   \[
   \frac{1}{2} m_2 V_2^2 = m_2 g h
   \]
   Solving for \( h \):
   \[
   h = \frac{V_2^2}{2g} = \frac{(1.305)^2}{2 \times 9.8} \approx 0.087 \, \text{m}
   \]

3. **Calculate \( \theta \):**
   Using trigonometry, \( \cos(\theta) = \frac{L - h}{L} \):
   \[
   \cos(\theta) = \frac{1.45 - 0.087}{1.45} \approx 0.940
   \]
   \[
   \theta = \cos^{-1}(0.940) \approx 19.948^\circ
   \]

### Conclusion:
The value \( \theta = 19.948^\circ \) is correct.

Would you like more details or have questions?

---

1. What principles of energy conservation are applied in pendulum motion?
2. How would the solution change if the string length \( L \) increased?
3. What if \( m_1 \) had a different velocity \( V_1 \)?
4. Can this method apply to inelastic collisions?
5. Why is conservation of momentum critical here?

**Tip:** Always verify units in physics problems for consistency and accuracy.

Verify if θ = 19.948° is correct in the given physics problem.

Explore a detailed solution for calculating the angle θ in a physics problem involving momentum and energy conservation. Given m1 = 0.27 kg, L = 1.45 m, m2 = 0.93 kg, and V1 = 4.5 m/s, determine if θ = 19.948° is correct. This problem applies trigonometry, kinetic energy, and potential energy principles, suitable for college physics students.

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