Major Types of Collisions

Why This Matters

Collisions are the bread and butter of momentum problems on the AP Physics C: Mechanics exam. You're being tested on your ability to recognize what's conserved (momentum always, kinetic energy sometimes) and apply that knowledge to predict final velocities, energy losses, and motion in multiple dimensions. The exam loves to give you a collision scenario and ask you to determine the type, calculate unknowns, or explain where energy went—so understanding the underlying physics is non-negotiable.

The key concepts here connect directly to conservation laws, impulse-momentum theorem, and energy transfer mechanisms. Every collision problem ultimately asks: Is this system isolated? What quantities are conserved? How do I set up my equations? Don't just memorize that "elastic means energy is conserved"—know why that matters for solving problems and how to identify collision types from given information. Master the reasoning, and the math follows naturally.

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Collisions Classified by Energy Conservation

The fundamental distinction between collision types comes down to what happens to kinetic energy. Momentum is always conserved in isolated systems, but kinetic energy tells you about the nature of the interaction itself.

Elastic Collisions

• Both momentum and kinetic energy are conserved—this is the defining characteristic and gives you two independent equations to solve for unknowns
• Objects rebound without permanent deformation—no energy is converted to heat, sound, or internal energy during the interaction
• Relative velocity reverses direction—for 1D elastic collisions, $v_{1f} - v_{2f} = -(v_{1i} - v_{2i})$, which provides a powerful shortcut for calculations

Inelastic Collisions

• Momentum is conserved, but kinetic energy is not—some kinetic energy transforms into heat, sound, or deformation energy
• Objects may deform during collision—real-world collisions are almost always inelastic to some degree
• Energy loss varies—the amount of kinetic energy lost depends on material properties and collision geometry

Perfectly Inelastic Collisions

• Objects stick together after collision—this is the defining feature, resulting in a single combined mass with common velocity
• Maximum kinetic energy loss occurs—more energy is lost than in any other collision type with the same initial conditions
• Simplest to solve mathematically—only one final velocity to find using $m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2)v_f$

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Collisions Classified by Geometry

The dimensionality of a collision determines how you set up your conservation equations. The physics doesn't change—only the number of component equations you need.

One-Dimensional Collisions

• Motion occurs along a single line—all velocities are either positive or negative along one axis
• Requires only scalar momentum conservation—$p_{i,total} = p_{f,total}$ in one direction simplifies calculations significantly
• Can be elastic or inelastic—the geometry doesn't determine energy conservation; the interaction does

Two-Dimensional Collisions

• Motion occurs in a plane—objects have velocity components in both $x$ and $y$ directions
• Momentum conserves independently in each direction—you get two equations: $p_{x,i} = p_{x,f}$ and $p_{y,i} = p_{y,f}$
• Vector analysis is essential—resolve initial velocities into components, apply conservation, then reconstruct final velocity magnitudes and angles

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Collisions Classified by Impact Geometry

How objects approach each other determines whether you're dealing with a simpler 1D problem or a more complex 2D analysis.

Head-On Collisions

• Objects collide directly along their line of motion—velocities are aligned (or anti-aligned) before impact
• Always one-dimensional—no perpendicular components of velocity exist, making this the simplest collision geometry
• Ideal for demonstrating conservation principles—most textbook examples use head-on collisions to isolate the physics from vector complexity

Glancing Collisions

• Objects collide at an angle—both objects change direction, creating a 2D momentum problem
• Requires component-by-component analysis—break velocities into $x$ and $y$ components before and after collision
• Often involves one object initially at rest—classic "billiard ball" problems where a moving ball strikes a stationary one off-center

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Conservation Laws in Collisions

These principles are the foundation of every collision problem. Understanding when and why they apply is more important than memorizing formulas.

Conservation of Momentum

• Total momentum of an isolated system is constant—$\vec{p}_{total,i} = \vec{p}_{total,f}$ always holds when no external impulse acts
• Applies to ALL collision types—elastic, inelastic, perfectly inelastic, 1D, 2D—momentum conservation is universal
• Momentum is a vector—direction matters; in 2D, you must conserve $p_x$ and $p_y$ separately

Conservation of Kinetic Energy

• Only applies to elastic collisions—$K_{total,i} = K_{total,f}$ gives you an additional equation beyond momentum conservation
• Enables solving for two unknowns—combined with momentum conservation, you can find both final velocities in elastic collisions
• Use the relative velocity shortcut—for 1D elastic collisions, the approach speed equals the separation speed, saving algebraic effort

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Quantifying Collision Elasticity

Coefficient of Restitution

• Defined as the ratio of relative speeds—$e = \frac{|v_{2f} - v_{1f}|}{|v_{1i} - v_{2i}|}$, measuring how "bouncy" a collision is
• Ranges from 0 to 1—$e = 1$ means perfectly elastic; $e = 0$ means perfectly inelastic (objects stick)
• Connects collision type to a single number—useful for real-world problems where collisions are partially elastic

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Quick Reference Table

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Self-Check Questions

1. Two objects collide and stick together. What type of collision is this, and what conservation law(s) can you apply to find the final velocity?

2. In a 2D glancing collision, why must you write separate momentum conservation equations for the $x$ and $y$ directions? What would go wrong if you only used one equation?

3. Compare elastic and inelastic collisions: both conserve momentum, so what additional information does knowing a collision is elastic give you when solving for unknowns?

4. A collision has a coefficient of restitution $e = 0.6$. Is this collision elastic, inelastic, or perfectly inelastic? How would you use this value in a calculation?

5. FRQ-style: Two pucks collide on a frictionless air table. Puck A (mass $m$) moves east at speed $v$ and strikes stationary Puck B (mass $2m$). After the collision, Puck A moves north. Explain why Puck B cannot also move north, and describe how you would find Puck B's velocity.