Thermodynamics and Elastic Collisions: Conservation of Momentum

#Collisions: A Deep Dive 💥

Let's break down collisions, a key topic in AP Physics 2. Remember, collisions involve objects interacting and exchanging energy and momentum. Whether it's a gentle bump or a massive crash, understanding these interactions is crucial.

#Conservation of Momentum

First things first: Momentum is King! 👑

• If the net external force on a system is zero (or negligibly small), the total momentum of the system is conserved. This means the total momentum before a collision equals the total momentum after the collision.
• Think of it like this: what goes in, must come out (in terms of total momentum).
• Momentum is a vector, so consider direction.

#Types of Collisions

Collisions come in two main flavors:

• Elastic Collisions 🏀
  • Both momentum and kinetic energy (KE) are conserved. Think of perfectly bouncy objects.
  • Objects bounce off each other without losing any energy to heat or sound.

* Example: Imagine billiard balls colliding; they mostly retain their kinetic energy.

• Inelastic Collisions 🧽
  • Momentum is conserved, but kinetic energy is NOT. Some KE is converted into other forms of energy (heat, sound, deformation).
  • Objects may stick together or deform.

* Example: A car crash; lots of energy is lost to deformation and heat.

#Elastic Collisions: The Details

#Key Principles

• Conservation of Kinetic Energy: The total kinetic energy of the system remains constant. This means: KEinitial = KEfinal
• Conservation of Momentum: The total momentum of the system remains constant. This means: pinitial = pfinal

#Solving Elastic Collision Problems

1. Identify: Determine if the collision is elastic. Look for keywords like "perfectly elastic" or if the problem states that KE is conserved.
2. Apply Conservation Laws:
   • Use the conservation of momentum equation: $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$
   • Use the conservation of kinetic energy equation: $1/2m_1v_{1i}^2 + 1/2m_2v_{2i}^2 = 1/2m_1v_{1f}^2 + 1/2m_2v_{2f}^2$
3. Solve: Solve the equations simultaneously for the unknowns (usually final velocities).
4. 2D Collisions: If the collision is in 2D, analyze momentum in x and y directions separately. Remember to use vector components.

#Example Problem Walkthrough

Let's revisit the example from the notes:

A cart with a mass of 2 kg is moving to the right at a velocity of 3 m/s. It collides with a stationary cart with a mass of 1 kg. After the collision, the first cart moves to the right at a velocity of 2 m/s, and the second cart moves to the right at a velocity of 1 m/s.

1. Classify the collision:

• Initial KE: KEi = 1/2 * (2 kg) * (3 m/s)^2 + 1/2 * (1 kg) * (0 m/s)^2 = 9 J
• Final KE: KEf = 1/2 * (2 kg) * (2 m/s)^2 + 1/2 * (1 kg) * (1 m/s)^2 = 4.5 J
• Since KEi ≠ KEf, this collision is inelastic (not elastic as stated in the original example).

2. Justify the selection of conservation of linear momentum:

• Even though kinetic energy is not conserved, the total linear momentum is always conserved when there is no external force.

3. Solve for missing variables:

• Initial momentum: pi = (2 kg * 3 m/s) + (1 kg * 0 m/s) = 6 kg*m/s
• Final momentum: pf = (2 kg * 2 m/s) + (1 kg * 1 m/s) = 5 kg*m/s
• The momentum is not conserved here as well, which means the final velocity given in the problem is incorrect.

4. Calculate their values:

• The example provided in the notes has incorrect calculations, but let's assume that the collision is elastic and solve the problem again.
• Initial momentum: pi = (2 kg * 3 m/s) + (1 kg * 0 m/s) = 6 kg*m/s
• Final momentum: pf = (2 kg * v1) + (1 kg * v2) = 6 kg*m/s
• Initial KE: KEi = 1/2 * (2 kg) * (3 m/s)^2 + 1/2 * (1 kg) * (0 m/s)^2 = 9 J
• Final KE: KEf = 1/2 * (2 kg) * (v1)^2 + 1/2 * (1 kg) * (v2)^2 = 9 J
• Solving these two equations simultaneously, we get v1 = 1 m/s and v2 = 4 m/s

#Inelastic Collisions: The Sticky Situation

#Key Principles

• Conservation of Momentum: The total momentum of the system is conserved. (pinitial = pfinal)
• Kinetic Energy is NOT Conserved: Some KE is lost to other forms of energy.

#Solving Inelastic Collision Problems

1. Identify: Determine if the collision is inelastic. Look for keywords like "stick together" or if the problem states that KE is not conserved.
2. Apply Conservation of Momentum:
   • Use the momentum equation: $m_1v_{1i} + m_2v_{2i} = (m_1 + m_2)v_f$ (for objects that stick together).
3. Solve: Solve for the unknown (usually final velocity).

#Connections to Thermodynamics

• In thermodynamics, we often assume collisions are elastic at the molecular level. This is a simplification, but it helps us understand how gases behave.
• The main point is to remember the difference between elastic and inelastic and when momentum is conserved.

#Final Exam Focus 🎯

• Prioritize: Focus on understanding the difference between elastic and inelastic collisions and when momentum is conserved.
• Practice: Solve various problems involving both types of collisions, including 2D collisions.
• Key Formulas:
  • Momentum: p = mv
  • Kinetic Energy: KE = 1/2mv^2
  • Conservation of Momentum: pinitial = pfinal
  • Conservation of Kinetic Energy (elastic): KEinitial = KEfinal
• Common Question Types:
  • Classifying collisions as elastic or inelastic.
  • Calculating final velocities after collisions.
  • Analyzing 2D collisions.

#Last-Minute Tips 🚀

• Time Management: Don't spend too much time on a single problem. If you're stuck, move on and come back later.
• Common Pitfalls: Be careful with vector directions. Remember that KE is a scalar, but momentum is a vector.
• Challenging Questions: Break down complex problems into smaller, manageable steps. Draw diagrams to visualize the situation.

#Practice Questions

Practice Question

Multiple Choice Questions

1. Two objects collide. Which of the following statements is always true? (A) The kinetic energy of the system is conserved. (B) The momentum of the system is conserved. (C) Both kinetic energy and momentum are conserved. (D) Neither kinetic energy nor momentum is conserved.

2. A ball of mass m moving with velocity v collides head-on with an identical ball at rest. If the collision is perfectly elastic, what is the velocity of the first ball after the collision? (A) 0 (B) v/2 (C) v (D) 2v

3. A 2 kg object moving at 3 m/s collides with a 1 kg object at rest. They stick together after the collision. What is their final velocity? (A) 1 m/s (B) 2 m/s (C) 3 m/s (D) 4 m/s

Free Response Question

A 0.5 kg cart is moving to the right on a frictionless track with a speed of 2 m/s. It collides with a 1.0 kg cart initially at rest. After the collision, the 0.5 kg cart moves to the left with a speed of 1 m/s.

(a) What is the velocity of the 1.0 kg cart after the collision? (b) Is the collision elastic or inelastic? Justify your answer. (c) How much kinetic energy is lost in the collision, if any?

Scoring Rubric

(a) 3 points

• 1 point: Correctly using conservation of momentum equation.
• 1 point: Correctly substituting values into the equation.
• 1 point: Correct answer with direction (1.5 m/s to the right)

(b) 3 points

• 1 point: Calculating initial kinetic energy (1 J)
• 1 point: Calculating final kinetic energy (1 J)
• 1 point: Correctly stating the collision is inelastic because KE is not conserved

(c) 2 points

• 1 point: Correctly calculating the total initial kinetic energy (1 J)
• 1 point: Correctly calculating the total final kinetic energy (0.875 J) and the energy lost (0.125 J)

Remember, you've got this! Stay calm, focus on the fundamentals, and you'll do great on the AP Physics 2 exam! 💪