#AP Physics 1: Collisions & Center of Mass - The Night Before 🚀

Hey there, future physicist! Let's get you prepped and confident for tomorrow's exam. We're going to break down collisions and center of mass, making sure everything clicks. Ready? Let's dive in!

#1. Collisions: Momentum is Key! 🎯

#1.1 What's a Collision?

A collision is when two or more objects interact, exerting forces on each other for a short period. Think of it like a quick high-five between objects. The most important thing to remember is that momentum is usually conserved during a collision.

#1.2 Types of Collisions

There are two main types of collisions:

• Elastic Collisions: Objects bounce off each other, and both momentum and kinetic energy are conserved. It's like a perfect billiard ball hit - no energy is lost to heat or sound.

• Inelastic Collisions: Objects either stick together or deform, and kinetic energy is not conserved (some energy is lost to heat, sound, etc.). Momentum is still conserved. Think of a car crash where the cars crumple.

*   **Completely Inelastic Collisions:** A special type of inelastic collision where objects stick together after colliding. 

#1.3 Visualizing Collisions

Inelastic Collision: Objects stick together after the collision, kinetic energy is not conserved.

Elastic Collision: Objects bounce off each other, both kinetic energy and momentum are conserved.

#1.4 Solving Collision Problems

1. Identify the Type: Determine if the collision is elastic or inelastic. This helps you know whether to conserve kinetic energy.

2. Conserve Momentum: Use the principle of conservation of momentum: $$p_{initial} = p_{final}$$ $$m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$$ Where:

   • $m_1$ and $m_2$ are the masses of the objects.
   • $v_{1i}$ and $v_{2i}$ are the initial velocities.
   • $v_{1f}$ and $v_{2f}$ are the final velocities.

3. Elastic Collisions (if applicable): If the collision is elastic, also conserve kinetic energy: $$KE_{initial} = KE_{final}$$ $$\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2$$

4. Solve: Use the equations to find the unknowns.

#1.5 Example: Inelastic Collision

Two carts of the same mass lie on a table. The first cart is moving and the second cart is at rest. At the end of the collision, the two carts were stuck together. What is the final speed of the system?

Solution:

• Initial momentum: $p_{initial} = m_1v_1 + m_2v_2$
• Final momentum: $p_{final} = (m_1 + m_2)v_{final}$
• Conservation of momentum: $p_{initial} = p_{final}$
• Solve for $v_{final}$

#2. Center of Mass: The Balance Point ⚖️

#2.1 What is Center of Mass?

The center of mass (COM) of a system is the point where the system's mass is evenly distributed. It's like the balancing point of an object. Imagine balancing a ruler on your finger; the point where it balances is its center of mass.

#2.2 Calculating Center of Mass

For a system of particles, the center of mass can be found using the following formula:

$$x_{cm} = \frac{m_1x_1 + m_2x_2 + ...}{m_1 + m_2 + ...}$$

Where:

• $x_{cm}$ is the x-coordinate of the center of mass.
• $m_1$, $m_2$,... are the masses of the particles.
• $x_1$, $x_2$,... are the x-coordinates of the particles.

You can use the same formula for the y-direction to find the y-coordinate of the center of mass.

#2.3 Center of Mass and Net Force

• Zero Net Force: If the net external force on a system is zero, the acceleration of the center of mass is zero. If the center of mass is moving, it will continue to move with the same velocity.

• Non-Zero Net Force: If there's a net external force, the center of mass will accelerate as if it were a single particle with the total mass of the system.

#2.4 Example: Center of Mass

A stick of length 1 meter is balanced on a fulcrum at its midpoint. A mass of 2 kg is attached to one end of the stick, and a mass of 3 kg is attached to the other end. What is the distance from the fulcrum to the center of mass of the system?

Solution:

• Use the center of mass formula: $x_{cm} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}$
• Plug in the values and solve for $x_{cm}$

#3. Final Exam Focus 🎯

#3.1 High-Value Topics

• Conservation of Momentum: This is huge! Understand when and how to apply it.
• Types of Collisions: Know the difference between elastic and inelastic collisions.
• Center of Mass: Be able to calculate it and understand its motion under different forces.

#3.2 Common Question Types

• Multiple Choice: Conceptual questions about momentum, energy conservation, and types of collisions.
• Free Response: Problems requiring you to calculate final velocities, kinetic energy changes, and center of mass positions.

#3.3 Last-Minute Tips

• Time Management: Don't get stuck on one problem. Move on and come back if you have time.
• Units: Always include units in your answers.
• Vector Nature: Remember that momentum and velocity are vectors. Consider direction!
• Draw Diagrams: Visualizing the problem can help you understand it better.
• Stay Calm: You've got this! Take deep breaths and approach each problem methodically.

#4. Practice Questions

Practice Question

#Multiple Choice Questions

1. Two objects collide. Which of the following is always conserved during the collision? (A) Kinetic energy (B) Momentum (C) Velocity (D) Both kinetic energy and momentum

2. A 2 kg ball moving at 3 m/s collides head-on with a 1 kg ball at rest. If the collision is perfectly inelastic, what is the final velocity of the combined mass? (A) 1 m/s (B) 2 m/s (C) 3 m/s (D) 4 m/s

3. A system consists of two particles with masses 2kg and 3kg. The 2 kg particle is located at (1,2) and the 3kg particle is located at (4, -1). What is the x coordinate of the center of mass of the system? (A) 2.8 (B) 3.0 (C) 3.2 (D) 3.4

#Free Response Question

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

(a) Calculate the velocity of cart B after the collision.

(b) Determine whether the collision is elastic or inelastic. Justify your answer.

(c) Calculate the center of mass of the system before the collision, given that cart A is at position x=0 and cart B is at position x=2m.

(d) If the collision was completely inelastic, calculate the final velocity of the combined carts.

Scoring Rubric:

(a) Velocity of cart B (3 points)

• 1 point: Correctly applying conservation of momentum: $m_Av_{Ai} + m_Bv_{Bi} = m_Av_{Af} + m_Bv_{Bf}$
• 1 point: Plugging in correct values: $(0.5)(2) + (1.0)(0) = (0.5)(-0.5) + (1.0)v_{Bf}$
• 1 point: Correctly solving for $v_{Bf}$: $v_{Bf} = 1.25 m/s$

(b) Type of Collision (3 points)

• 1 point: Calculating initial kinetic energy: $KE_i = \frac{1}{2}m_Av_{Ai}^2 + \frac{1}{2}m_Bv_{Bi}^2 = \frac{1}{2}(0.5)(2)^2 + 0 = 1 J$
• 1 point: Calculating final kinetic energy: $KE_f = \frac{1}{2}m_Av_{Af}^2 + \frac{1}{2}m_Bv_{Bf}^2 = \frac{1}{2}(0.5)(-0.5)^2 + \frac{1}{2}(1)(1.25)^2 = 0.8125 J$
• 1 point: Correctly stating the collision is inelastic because $KE_i ≠ KE_f$

(c) Center of Mass Before Collision (2 points)

• 1 point: Using the center of mass formula: $x_{cm} = \frac{m_Ax_A + m_Bx_B}{m_A + m_B}$
• 1 point: Correctly calculating $x_{cm} = \frac{(0.5)(0) + (1.0)(2)}{0.5 + 1.0} = 1.33 m$

(d) Final Velocity for Completely Inelastic Collision (2 points)

• 1 point: Using conservation of momentum for completely inelastic collision: $m_Av_{Ai} + m_Bv_{Bi} = (m_A + m_B)v_f$
• 1 point: Correctly solving for $v_f$: $v_f = \frac{(0.5)(2) + (1.0)(0)}{0.5 + 1.0} = 0.67 m/s$

Alright, you've got this! Go ace that exam! 💪