AP Physics – Elastic vs Inelastic Collisions

AP Physics – Elastic vs Inelastic Collisions

AP Physics: Elastic vs. Inelastic Collisions – Exam-Ready Study Guide

What This Is

Elastic and inelastic collisions describe how objects interact when they collide, focusing on whether kinetic energy (KE) is conserved. This topic is high-yield on the AP Physics exam because it tests your understanding of conservation laws (momentum and energy), problem-solving with systems, and real-world applications (e.g., car crashes, particle physics, sports). Example: A tennis ball bouncing off a racket is nearly elastic (KE is mostly conserved), while a lump of clay sticking to a wall is perfectly inelastic (KE is lost as heat/sound).

Key Terms & Concepts

• Collision: An interaction between two or more objects where forces act over a short time, changing their motion.
• Momentum (p): p = mv (mass × velocity). Always conserved in collisions if no external forces act.
• Kinetic Energy (KE): KE = ½mv². Not always conserved in collisions.
• Elastic Collision: A collision where both momentum and kinetic energy are conserved. Objects bounce off each other without permanent deformation (e.g., billiard balls, ideal gas molecules).
• Inelastic Collision: A collision where momentum is conserved, but kinetic energy is not. Some KE is lost as heat, sound, or deformation (e.g., car crashes, clay sticking to a wall).
• Perfectly Inelastic Collision: A special case of inelastic collisions where objects stick together after colliding (e.g., two train cars coupling). Maximum KE loss occurs here.
• Coefficient of Restitution (e): A measure of "bounciness" (0-e-1). e = 1 = elastic, e = 0 = perfectly inelastic.
• Conservation of Momentum: m?v + m?v = m?v?f + m?v?f (initial momentum = final momentum).
• Conservation of KE (Elastic Only): ½m?v² + ½m?v² = ½m?v?f² + ½m?v?f².
• Relative Velocity (Elastic Collisions): In 1D elastic collisions, v – v = –(v?f – v?f) (relative velocity reverses direction).
• Center of Mass (COM): The average position of mass in a system. COM velocity is constant if no external forces act (useful for analyzing collisions).

Step-by-Step: Solving Collision Problems

1. Identify the type of collision (elastic, inelastic, or perfectly inelastic) from the problem description.
2. Draw a before/after diagram showing masses, velocities, and directions.
3. Write the conservation of momentum equation (always valid): m?v + m?v = m?v?f + m?v?f (or m?v + m?v = (m? + m?)vf for perfectly inelastic).
4. For elastic collisions only, write the conservation of KE equation: ½m?v² + ½m?v² = ½m?v?f² + ½m?v?f².
5. Solve the system of equations (2 equations for elastic, 1 for inelastic). For 1D elastic collisions, use the relative velocity shortcut: v?f = [(m? – m?)/(m? + m?)]v + [2m?/(m? + m?)]v v?f = [2m?/(m? + m?)]v + [(m? – m?)/(m? + m?)]v.
6. Check units and reasonableness (e.g., final velocities should make sense for the collision type).

Common Mistakes

• Mistake: Assuming kinetic energy is always conserved. Correction: KE is only conserved in elastic collisions. In inelastic collisions, some KE is lost (e.g., as heat or sound). Why? Energy can transform into other forms, but momentum is always conserved in isolated systems.

• Mistake: Forgetting to account for direction (signs) in momentum equations. Correction: Assign a coordinate system (e.g., right = +, left = –) and stick to it. Why? Momentum is a vector; direction matters!

• Mistake: Using the elastic collision KE equation for inelastic collisions. Correction: Only use KE conservation for elastic collisions. For inelastic collisions, use momentum conservation and maybe the coefficient of restitution (if given). Why? KE isn’t conserved in inelastic collisions.

• Mistake: Misapplying the perfectly inelastic formula (objects sticking together). Correction: For perfectly inelastic collisions, the final velocity is vf = (m?v + m?v)/(m? + m?). Why? The objects move as one mass after collision.

• Mistake: Ignoring external forces (e.g., friction, gravity). Correction: Momentum is only conserved if no external forces act during the collision. Why? External forces can change the system’s total momentum.

AP Exam Insights

• Tricky Distinction: The AP exam loves testing whether a collision is elastic or inelastic. Key clue: If objects stick together-perfectly inelastic. If they bounce and KE is conserved-elastic.
• FRQ Favorite: You’ll often get a 2-object collision problem where you must:
• Calculate final velocities using momentum conservation.
• Determine if KE is conserved (and calculate KE loss if not).
• Explain why the collision is elastic/inelastic (e.g., "KE is conserved because no energy is lost to deformation").
• Multiple-Choice Traps:
• Answer choices mixing momentum and KE conservation (e.g., "Momentum is conserved but KE is not" for inelastic collisions).
• Problems with "nearly elastic" collisions (e.g., a superball bounce with e-0.9). Solution: Treat as elastic unless told otherwise.
• Real-World Context: Expect questions about car crashes, sports (e.g., baseball/bat collisions), or particle physics (e.g., alpha particles scattering off nuclei).

Quick Check Questions

1. Multiple Choice: A 2 kg cart moving at 3 m/s collides with and sticks to a stationary 1 kg cart. What is their final velocity? a) 1 m/s b) 2 m/s c) 3 m/s d) 4 m/s Answer: b) 2 m/s. Explanation: Use perfectly inelastic collision formula: vf = (2×3 + 1×0)/(2+1) = 6/3 = 2 m/s.

2. Short FRQ: A 0.5 kg ball moving at 4 m/s collides elastically with a stationary 1 kg ball. What are their final velocities? Answer:

3. v?f = [(0.5 – 1)/(0.5 + 1)]×4 + [2×1/(0.5 + 1)]×0 = –4/3 m/s (reverses direction).

4. v?f = [2×0.5/(0.5 + 1)]×4 + [(1 – 0.5)/(0.5 + 1)]×0 = 8/3 m/s. Explanation: Use the 1D elastic collision formulas.

5. Multiple Choice: In which collision is kinetic energy not conserved? a) Two billiard balls colliding b) A tennis ball bouncing off a racket c) A bullet embedding in a block of wood d) Two gas molecules colliding Answer: c) A bullet embedding in a block of wood. Explanation: This is a perfectly inelastic collision (objects stick together, KE is lost).

Last-Minute Cram Sheet

1. Momentum is always conserved in collisions (if no external forces). KE is only conserved in elastic collisions.
2. Elastic collision: Objects bounce, KE conserved. Inelastic collision: Objects deform, KE lost.
3. Perfectly inelastic: Objects stick together. Use vf = (m?v + m?v)/(m? + m?).
4. 1D elastic collision shortcut: Relative velocity reverses: v – v = –(v?f – v?f).
5. Coefficient of restitution (e): e = 1 (elastic), e = 0 (perfectly inelastic).
6. Don’t assume KE is conserved! Check the problem for "elastic" or "inelastic."
7. Direction matters! Assign +/– signs to velocities.
8. KE loss in inelastic collisions = Initial KE – Final KE.
9. Center of mass velocity is constant if no external forces act.
10. Real-world examples: Elastic = billiard balls, inelastic = car crashes, perfectly inelastic = train cars coupling.