#AP Physics C: Mechanics - Momentum Study Guide

Hey there, future physics pro! Let's get you prepped for the AP exam with a deep dive into momentum. This guide is designed to be your go-to resource, especially the night before the exam. We'll break down the key concepts, highlight important connections, and make sure you're feeling confident and ready to ace it! Let's get started!

#1. Linear Momentum: The Basics

#1.1. What is Linear Momentum?

• Definition: Linear momentum ($\vec{p}$) is a measure of how much "oomph" an object has in its motion. It's the product of an object's mass ($m$) and its velocity ($\vec{v}$). 📏

  $$\vec{p} = m \vec{v}$$

• It's a vector, meaning it has both magnitude and direction.

• Think of it this way: a truck moving slowly has more momentum than a baseball moving fast, because of its larger mass.

#1.2. Direction of Momentum

• Direction: The direction of momentum is always the same as the direction of the object's velocity. 🧭

• If an object is moving to the right, its momentum vector points to the right. If it changes direction, the momentum vector changes direction too.

• Opposite Velocities: Objects with the same mass but opposite velocities have equal magnitudes of momentum, but in opposite directions.

#1.3. Momentum in Collisions and Explosions 💥

• Collisions: These involve brief, intense interactions between objects. The forces involved are large, but the time of interaction is short.

  • During collisions, the forces between objects are much greater than any external forces, so we can often ignore external forces during the collision.

• Explosions: These involve internal forces pushing objects apart. Objects initially at rest or close together gain momentum and move away from each other.

• Key Concept: We often use the object model to simplify analysis. We focus on the initial and final states of the objects, ignoring the complex forces during the interaction.

• Conservation of Momentum: This principle allows us to determine final velocities based on initial conditions, even if we don't know the exact forces involved. This is a huge time-saver!

Practice Question

Multiple Choice Questions

1. A 2 kg object is moving with a velocity of 5 m/s to the right. What is the momentum of the object? (A) 2.5 kg m/s to the right (B) 10 kg m/s to the right (C) 2.5 kg m/s to the left (D) 10 kg m/s to the left

2. Two objects, one with a mass of 1 kg and another with a mass of 2 kg, are moving with the same velocity. How do their momenta compare? (A) The 1 kg object has twice the momentum of the 2 kg object. (B) The 2 kg object has twice the momentum of the 1 kg object. (C) Both objects have the same momentum. (D) The momentum of the objects cannot be compared without knowing the velocity

Free Response Question

Two blocks are on a frictionless horizontal surface. Block A has a mass of 2 kg and is moving to the right with a velocity of 5 m/s. Block B has a mass of 3 kg and is initially at rest. The two blocks collide and stick together.

(a) What is the initial momentum of block A? (b) What is the initial momentum of block B? (c) What is the total initial momentum of the system? (d) What is the final velocity of the two blocks after the collision?

Answer Key

Multiple Choice Questions

1. (B) 10 kg m/s to the right
2. (B) The 2 kg object has twice the momentum of the 1 kg object.

Free Response Question

(a) Initial momentum of block A: $p_A = m_A v_A = (2 \text{ kg})(5 \text{ m/s}) = 10 \text{ kg m/s}$ to the right (2 points) (b) Initial momentum of block B: $p_B = m_B v_B = (3 \text{ kg})(0 \text{ m/s}) = 0 \text{ kg m/s}$ (1 point) (c) Total initial momentum: $p_{total} = p_A + p_B = 10 \text{ kg m/s} + 0 \text{ kg m/s} = 10 \text{ kg m/s}$ to the right (1 point) (d) Final velocity of the two blocks: Conservation of momentum: $p_{initial} = p_{final}$. $10 \text{ kg m/s} = (m_A + m_B) v_f$. $10 \text{ kg m/s} = (2 \text{ kg} + 3 \text{ kg}) v_f$. $v_f = 2 \text{ m/s}$ to the right (3 points)

#2. Impulse and the Impulse-Momentum Theorem

#2.1. What is Impulse?

• Definition: Impulse ($\vec{J}$) is the change in momentum of an object. It's also equal to the net force ($\vec{F}$) acting on an object multiplied by the time interval ($\Delta t$) over which the force acts.

$$\vec{J} = \vec{F} \Delta t$$

• Impulse is also a vector, and its direction is the same as the direction of the net force.

• Think of impulse as the "push" that changes an object's momentum. A larger force or a longer time leads to a larger impulse.

#2.2. The Impulse-Momentum Theorem

• Theorem: The impulse on an object is equal to the change in its momentum.

$$\vec{J} = \Delta \vec{p} = m \vec{v}_f - m \vec{v}_i$$

• This theorem connects force, time, and changes in momentum. It's a powerful tool for analyzing collisions and other interactions.

• The impulse-momentum theorem is derived from Newton's second law of motion.

#2.3. Applications of Impulse

• Reducing Impact Force: In situations like car crashes, increasing the time of impact (e.g., with airbags) reduces the force experienced by the occupants, because the impulse remains the same.

• Increasing Momentum Change: In sports, a longer contact time (e.g., following through on a swing) increases the impulse and thus the change in momentum of the ball.

Practice Question

Multiple Choice Questions

1. A 1 kg ball is dropped from a height and hits the ground with a velocity of -5 m/s. It bounces back up with a velocity of +4 m/s. What is the magnitude of the impulse on the ball? (A) 1 kg m/s (B) 4 kg m/s (C) 5 kg m/s (D) 9 kg m/s

2. A constant force of 10 N is applied to a 2 kg object for 3 seconds. What is the impulse on the object? (A) 5 N s (B) 15 N s (C) 30 N s (D) 60 N s

Free Response Question

A 0.5 kg baseball is thrown with an initial velocity of 30 m/s to the right. It is struck by a bat, and its final velocity is 40 m/s to the left. The bat is in contact with the ball for 0.002 seconds.

(a) What is the initial momentum of the baseball? (b) What is the final momentum of the baseball? (c) What is the impulse on the baseball? (d) What is the average force exerted by the bat on the baseball?

Answer Key

Multiple Choice Questions

1. (D) 9 kg m/s
2. (C) 30 N s

Free Response Question

(a) Initial momentum of the baseball: $p_i = m v_i = (0.5 \text{ kg})(30 \text{ m/s}) = 15 \text{ kg m/s}$ to the right (1 point) (b) Final momentum of the baseball: $p_f = m v_f = (0.5 \text{ kg})(-40 \text{ m/s}) = -20 \text{ kg m/s}$ to the left (1 point) (c) Impulse on the baseball: $J = \Delta p = p_f - p_i = -20 \text{ kg m/s} - 15 \text{ kg m/s} = -35 \text{ kg m/s}$ to the left (2 points) (d) Average force exerted by the bat: $J = F \Delta t$. $-35 \text{ kg m/s} = F (0.002 \text{ s})$. $F = -17500 \text{ N}$ to the left (3 points)

#3. Conservation of Linear Momentum

#3.1. The Principle of Conservation of Momentum

• Principle: In a closed system (no external forces), the total momentum remains constant. That is, the total momentum before an interaction equals the total momentum after the interaction.

$$\vec{p}_{initial} = \vec{p}_{final}$$

• This is a fundamental law of physics, and it's incredibly useful for solving problems involving collisions and explosions.

• It applies to both elastic and inelastic collisions, as long as the system is closed.

Conservation of momentum is a high-value topic on the AP exam. Expect to see multiple questions that require applying this principle in various scenarios.

#3.2. Types of Collisions

• Elastic Collisions: Kinetic energy is conserved. Objects bounce off each other without losing energy to heat or deformation. Momentum is always conserved in both elastic and inelastic collisions.

• Inelastic Collisions: Kinetic energy is not conserved. Some kinetic energy is converted into other forms of energy, such as heat or sound. Objects may stick together after the collision.

• Perfectly Inelastic Collisions: A special type of inelastic collision where objects stick together after colliding. The final velocity of the combined mass can be calculated using conservation of momentum.

#3.3. Applying Conservation of Momentum

• Problem-Solving Strategy:

  1. Identify the system and ensure it's closed (no external forces).
  2. Write down the initial momentum of all objects in the system.
  3. Write down the final momentum of all objects in the system.
  4. Set the total initial momentum equal to the total final momentum. $\vec{p}_{initial} = \vec{p}_{final}$
  5. Solve for the unknown quantities.

• Complex Scenarios: Conservation of momentum can be applied in two or three dimensions, but you'll need to consider the vector components of momentum.

Practice Question

Multiple Choice Questions

1. Two carts of equal mass move towards each other on a frictionless track. Cart A has a velocity of 2 m/s to the right, and cart B has a velocity of 3 m/s to the left. If they collide and stick together, what is the final velocity of the combined carts? (A) 0.5 m/s to the left (B) 0.5 m/s to the right (C) 1 m/s to the left (D) 1 m/s to the right

2. A 2 kg object moving at 5 m/s collides elastically with a 1 kg object at rest. Which of the following is true after the collision? (A) The total kinetic energy of the system is less than before the collision. (B) The total momentum of the system is less than before the collision. (C) The total kinetic energy and momentum of the system are conserved. (D) The total momentum of the system is conserved but not the kinetic energy.

Free Response Question

A 4 kg block is moving to the right with a velocity of 6 m/s on a frictionless horizontal surface. It collides with a 2 kg block that is initially at rest. After the collision, the 4 kg block continues to move to the right with a velocity of 2 m/s.

(a) What is the initial momentum of the 4 kg block? (b) What is the initial momentum of the 2 kg block? (c) What is the total initial momentum of the system? (d) What is the final momentum of the 4 kg block? (e) What is the final velocity of the 2 kg block? (f) Is this collision elastic or inelastic? Justify your answer.

Answer Key

Multiple Choice Questions

1. (A) 0.5 m/s to the left
2. (C) The total kinetic energy and momentum of the system are conserved.

Free Response Question

(a) Initial momentum of the 4 kg block: $p_{4i} = m_4 v_{4i} = (4 \text{ kg})(6 \text{ m/s}) = 24 \text{ kg m/s}$ to the right (1 point) (b) Initial momentum of the 2 kg block: $p_{2i} = m_2 v_{2i} = (2 \text{ kg})(0 \text{ m/s}) = 0 \text{ kg m/s}$ (1 point) (c) Total initial momentum: $p_{total_i} = p_{4i} + p_{2i} = 24 \text{ kg m/s} + 0 \text{ kg m/s} = 24 \text{ kg m/s}$ to the right (1 point) (d) Final momentum of the 4 kg block: $p_{4f} = m_4 v_{4f} = (4 \text{ kg})(2 \text{ m/s}) = 8 \text{ kg m/s}$ to the right (1 point) (e) Final velocity of the 2 kg block: $p_{total_i} = p_{total_f}$. $24 \text{ kg m/s} = 8 \text{ kg m/s} + p_{2f}$. $p_{2f} = 16 \text{ kg m/s}$. $v_{2f} = p_{2f} / m_2 = 16 \text{ kg m/s} / 2 \text{ kg} = 8 \text{ m/s}$ to the right (2 points) (f) This collision is inelastic because kinetic energy is not conserved. Initial kinetic energy is $KE_i = 1/2 (4)(6^2) = 72 J$. Final kinetic energy is $KE_f = 1/2(4)(2^2) + 1/2(2)(8^2) = 4 + 64 = 68 J$. Since KE is not conserved, the collision is inelastic. (2 points)

#4. Final Exam Focus

#4.1. High-Priority Topics

• Conservation of Momentum: This is a must-know. Expect to see it in various forms, including collisions, explosions, and multi-dimensional scenarios.
• Impulse-Momentum Theorem: Understand how force, time, and momentum changes are related. Be ready to use it to solve problems involving impacts and changes in motion.
• Elastic vs. Inelastic Collisions: Know the differences and how to apply conservation of momentum and kinetic energy (for elastic collisions).

#4.2. Common Question Types

• Multiple Choice Questions: Often test your understanding of definitions, relationships, and basic problem-solving skills.
• Free Response Questions: Usually involve applying conservation of momentum and/or the impulse-momentum theorem in more complex, multi-step problems.
  • Pay close attention to the direction of vectors and be sure to include units in your answers.

#4.3. Last-Minute Tips

• Time Management: Don't spend too long on a single question. If you're stuck, move on and come back to it later.
• Common Pitfalls:
  • Forgetting that momentum is a vector quantity.
  • Confusing impulse with force.
  • Not checking if the system is closed before applying conservation of momentum.
  • Misidentifying elastic vs. inelastic collisions.
• Strategies:
  • Always draw diagrams to visualize the situation.
  • Write down the given information and what you're trying to find.
  • Use the problem-solving strategies we've discussed in this guide.
  • Double-check your units and calculations.

Good luck, you've got this! 💪