Solved Examples of Elastic Collisions in One and Two Dimensions

📚 Quick Study Guide

• 📏 Elastic Collision Definition: A collision where the total kinetic energy of the system is conserved.
• 💡 Key Conditions: No energy is lost to heat, sound, or deformation.
• ⚖️ Conservation Laws: Both momentum and kinetic energy are conserved.
• ➗ One-Dimensional Collisions:
  • ➡️ Conservation of Momentum: $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$
  • 📈 Conservation of Kinetic Energy: $\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2$
• 📐 Two-Dimensional Collisions:
  • ➡️ Conservation of Momentum (x-component): $m_1v_{1ix} + m_2v_{2ix} = m_1v_{1fx} + m_2v_{2fx}$
  • ⬆️ Conservation of Momentum (y-component): $m_1v_{1iy} + m_2v_{2iy} = m_1v_{1fy} + m_2v_{2fy}$
• 🧮 Coefficient of Restitution (e): $e = \frac{|v_{2f} - v_{1f}|}{|v_{1i} - v_{2i}|} = 1$ (for perfectly elastic collisions)

🧪 Practice Quiz

1. Question 1: In a perfectly elastic collision, which of the following quantities is/are conserved?
   1. A. Only momentum
   2. B. Only kinetic energy
   3. C. Both momentum and kinetic energy
   4. D. Neither momentum nor kinetic energy
2. Question 2: Two identical billiard balls collide elastically. If ball 1 is initially moving with velocity $v$ and ball 2 is at rest, what is the velocity of ball 2 after the collision?
   1. A. 0
   2. B. $v/2$
   3. C. $v$
   4. D. $2v$
3. Question 3: What is the coefficient of restitution for a perfectly elastic collision?
   1. A. 0
   2. B. 0.5
   3. C. 1
   4. D. 2
4. Question 4: A 2 kg ball moving at 3 m/s collides elastically with a 1 kg ball at rest. What is the approximate velocity of the 1 kg ball after the collision?
   1. A. 1 m/s
   2. B. 2 m/s
   3. C. 4 m/s
   4. D. 6 m/s
5. Question 5: In a two-dimensional elastic collision, what is conserved along each axis?
   1. A. Only kinetic energy
   2. B. Only momentum
   3. C. Both momentum and kinetic energy
   4. D. Neither momentum nor kinetic energy
6. Question 6: A ball collides elastically with a stationary wall. What happens to the ball's kinetic energy?
   1. A. Increases
   2. B. Decreases
   3. C. Remains the same
   4. D. Becomes zero
7. Question 7: Two objects with equal mass undergo an elastic head-on collision. If one object was initially at rest, what is the relative velocity of the objects after the collision?
   1. A. 0
   2. B. Equal to the initial velocity of the moving object
   3. C. Half the initial velocity of the moving object
   4. D. Twice the initial velocity of the moving object