#AP Physics C: Mechanics - Momentum & Collisions 🚀

Hey! Let's get you totally prepped for the exam with a super-focused review of momentum and collisions. We'll break down the key concepts, hit the most important points, and make sure you're ready to tackle any question they throw at you. Let's do this!

#Conservation of Linear Momentum

#What is Momentum? 🤔

• Momentum ($\vec{p}$) is a measure of an object's motion. It's a vector, so it has both magnitude and direction.
• Formula: $\vec{p} = m\vec{v}$ (mass times velocity)

#The Big Idea: Conservation of Linear Momentum 💡

• Key Concept: In a closed system (no external forces), the total momentum remains constant. It's like a cosmic bank account – what you start with is what you end with.

• Important: This applies to the total momentum of the system, not necessarily individual objects within the system.

#Visualizing Conservation

Caption: The total momentum before an event (like a collision) equals the total momentum after the event.

#Why Does Momentum Conserve? 🤔

• It's all thanks to Newton's Third Law: For every action, there's an equal and opposite reaction. This means that when objects interact, the forces they exert on each other are equal and opposite, leading to no net change in momentum for the system.
• Impulse: Impulse ($\vec{J}$) is the change in momentum. If there's no external force, there's no impulse, and thus no change in momentum. $\vec{J} = \Delta\vec{p} = \vec{F}\Delta t$

• Think of momentum as the "oomph" of a moving object. A heavier train or faster train has more "oomph" or momentum.
• If no external force acts on the train system, its total momentum stays the same, even if the train cars bump into each other.

#Collisions

#What's a Collision? 💥

• A collision is any interaction between objects where they exert forces on each other for a short period. Think billiard balls, cars crashing, or even a rocket launching.
• The forces within the system don't change the system's total momentum, because they're internal forces. Only external forces can change the total momentum.

#Types of Collisions

#Elastic Collisions 🏓

• Key Features:
  • Momentum is conserved.
  • Kinetic energy is conserved (no energy lost to heat or deformation).
  • Think of billiard balls colliding – they bounce off each other with no loss of energy.

#Inelastic Collisions 🚗

• Key Features:
  • Momentum is conserved.
  • Kinetic energy is NOT conserved (some energy is lost, often as heat or deformation).
  • Think of a car crash – the cars deform, and some energy is converted to heat and sound.
  • Perfectly Inelastic Collisions: The objects stick together after the collision. This results in the maximum loss of kinetic energy.

#Visualizing Collisions

Caption: A visual representation of a collision, showing the initial and final states of the objects involved.

• Elastic: Think bouncy balls – they keep their energy when they bounce.
• Inelastic: Think of play dough – when it hits something, it sticks and loses energy.

• Identify the System: Clearly define what objects are part of your system. This helps determine if momentum is conserved.
• Vector Nature: Remember that momentum is a vector. Pay attention to directions (positive and negative).
• Conservation Equation: $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$. Use this to set up your equations.
• Elastic vs. Inelastic: Determine if kinetic energy is conserved. If it's not, you can't use kinetic energy conservation.

#Final Exam Focus 🎯

#High-Priority Topics

• Conservation of Linear Momentum: This is HUGE. Make sure you understand the concept and when it applies.
• Elastic vs. Inelastic Collisions: Know the difference and how to approach problems involving each type.
• Impulse: Understand how impulse relates to changes in momentum.

#Common Question Types

• Collision Problems: You'll likely see problems involving two or more objects colliding. Be ready to use conservation of momentum.
• Explosions: These are just reverse collisions. Momentum is still conserved.
• Conceptual Questions: Expect questions that test your understanding of the underlying principles, not just calculations.

#Last-Minute Tips

• Time Management: Don't spend too long on one question. If you're stuck, move on and come back later.
• Units: Always include units in your answers. It's an easy way to lose points if you forget.
• Show Your Work: Even if you make a mistake, showing your work can earn you partial credit.
• Stay Calm: You've got this! Take deep breaths and approach each question methodically.

#Practice Questions

Practice Question

#Multiple Choice Questions

Question 1:

Two objects of equal mass move with equal speeds in opposite directions. They collide head-on and stick together. What is the velocity of the combined object after the collision?

(A) Zero (B) Equal to the initial speed of either object (C) Twice the initial speed of either object (D) Half the initial speed of either object

Question 2:

A 2 kg cart moving at 3 m/s collides with a 1 kg cart at rest. If the collision is perfectly inelastic, what is the final velocity of the combined carts?

(A) 1 m/s (B) 2 m/s (C) 3 m/s (D) 4 m/s

#Free Response Question

Context:

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

(a) What is the velocity of the 1 kg cart after the collision?

(b) Is this collision elastic or inelastic? Justify your answer.

(c) Calculate the change in kinetic energy of the system.

Scoring Rubric:

(a) (4 points)

• 1 point: Correctly using conservation of momentum principle.
• 1 point: Correctly setting up the equation with initial and final momenta.
• 1 point: Correctly substituting the values.
• 1 point: Correctly solving for the final velocity of the 1 kg cart.

(b) (3 points)

• 1 point: Correctly stating whether the collision is elastic or inelastic
• 2 points: Justifying the answer by checking for conservation of kinetic energy.

(c) (3 points)

• 1 point: Correctly calculating the initial kinetic energy of the system.
• 1 point: Correctly calculating the final kinetic energy of the system.
• 1 point: Correctly calculating the change in kinetic energy.

#Answers

MCQ 1: (A) Zero

MCQ 2: (B) 2 m/s

FRQ:

(a) Using conservation of momentum:

$m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$

$(0.5 kg)(2 m/s) + (1 kg)(0 m/s) = (0.5 kg)(-1 m/s) + (1 kg)v_{2f}$

$1 = -0.5 + v_{2f}$

$v_{2f} = 1.5 m/s$

(b) To determine if the collision is elastic, we need to check if kinetic energy is conserved:

Initial KE = $\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}(0.5)(2)^2 + 0 = 1 J$

Final KE = $\frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2 = \frac{1}{2}(0.5)(-1)^2 + \frac{1}{2}(1)(1.5)^2 = 0.25 + 1.125 = 1.375 J$

Since initial KE ≠ final KE, this collision is inelastic.

(c) Change in KE = Final KE - Initial KE = 1.375 J - 1 J = 0.375 J

#Additional Practice Problems

Practice Question

Question 1:

Taken fromxa0College Board

(For part 1 you'll need the answer for part a of this question, which is -2.03 N*s)

Answer: Impulse is equal to the change in momentum, and in this scenario mass is not changing so we can look at this problem with initial and final velocity. Remember that velocity is a vector, so positives and negatives represent direction!

Question 2:

Answer:

Use conservation of momentum to calculate the speed of cart 2. Then, you should recall that kinetic energy is conserved in elastic collisions.