Introduction
Collisions involve brief, strong interactions between objects. Momentum is always conserved during collisions; kinetic energy may or may not be conserved. Classification by energy behavior leads to three collision types: elastic (KE conserved), inelastic (KE not conserved), and completely inelastic (maximum KE loss, objects stick).
Elastic Collisions
In elastic collisions, both momentum and kinetic energy are conserved. For two bodies: m1v1 + m2v2 = m1v1' + m2v2' (momentum) and 1/2m1v₲ + 1/2m2v22 = 1/2m1v1'2 + 1/2m2v2'2 (energy). Special case - one-dimensional elastic collision with target at rest: v1' = (m1-m2)v1/(m1+m2), v2' = 2m1v1/(m1+m2). Equal masses exchange velocities.
Inelastic Collisions
In inelastic collisions, momentum is conserved but kinetic energy is not. Some KE converts to other forms: heat, sound, deformation energy. Coefficient of restitution e = (v2'-v1')/(v1-v2) where 0 < e < 1 for inelastic collisions. e = 1 is elastic, e = 0 is completely inelastic. Real collisions have intermediate e values.
\nIntroduction
Collisions involve brief, strong interactions between objects. Momentum is always conserved during collisions; kinetic energy may or may not be conserved. Classification by energy behavior leads to three collision types: elastic (KE conserved), inelastic (KE not conserved), and completely inelastic (maximum KE loss, objects stick).
Elastic Collisions
In elastic collisions, both momentum and kinetic energy are conserved. For two bodies: m1v1 + m2v2 = m1v1' + m2v2' (momentum) and 1/2m1v₲ + 1/2m2v22 = 1/2m1v1'2 + 1/2m2v2'2 (energy). Special case - one-dimensional elastic collision with target at rest: v1' = (m1-m2)v1/(m1+m2), v2' = 2m1v1/(m1+m2). Equal masses exchange velocities.
Inelastic Collisions
In inelastic collisions, momentum is conserved but kinetic energy is not. Some KE converts to other forms: heat, sound, deformation energy. Coefficient of restitution e = (v2'-v1')/(v1-v2) where 0 < e < 1 for inelastic collisions. e = 1 is elastic, e = 0 is completely inelastic. Real collisions have intermediate e values.
\nCompletely Inelastic Collisions
In completely inelastic collisions, objects stick together after collision, moving with common velocity. Momentum conservation: m1v1 + m2v2 = (m1+m2)v'. Final velocity: v' = (m1v1 + m2v2)/(m1+m2). Maximum kinetic energy is lost consistent with momentum conservation. Energy loss: ΔK = 1/2μ(v1-v2)2 where μ = m1m2/(m1+m2) is reduced mass.
Two-Dimensional Collisions
In 2D or 3D, momentum conservation applies separately to each component. For elastic collisions, three equations (momentum in x, momentum in y, energy) can determine four unknowns (two velocity components for each outgoing particle). One parameter remains free - typically the scattering angle. For inelastic collisions, energy equation is replaced by the coefficient of restitution relation.
\nApplications
Applications: Nuclear reactors (neutron moderation requires elastic collisions with light nuclei); Billiards (nearly elastic collisions); Crash testing (inelastic - energy absorption); Ballistic pendulum (completely inelastic - measures projectile velocity); Particle accelerators (scattering experiments); Sports (ball collisions, follow-through technique).
Solved Example: Elastic Collision of Two Balls
Ball A (2 kg) moving at 3 m/s collides elastically with stationary Ball B (1 kg) on frictionless surface. Find final velocities. Solution: Momentum: 2×3 + 0 = 2v_A' + 1×v_B' → 6 = 2v_A' + v_B'. Energy: 1/2×2×9 + 0 = 1/2×2×v_A'2 + 1/2×1×v_B'2 → 9 = v_A'2 + 0.5v_B'2. From momentum: v_B' = 6 - 2v_A'. Substitute into energy: 9 = v_A'2 + 0.5(6-2v_A')2 = v_A'2 + 0.5(36 - 24v_A' + 4v_A'2) = v_A'2 + 18 - 12v_A' + 2v_A'2 = 3v_A'2 - 12v_A' + 18. So 3v_A'2 - 12v_A' + 9 = 0 → v_A'2 - 4v_A' + 3 = 0 → (v_A'-1)(v_A'-3) = 0. v_A' = 3 m/s means no collision (initial condition), so v_A' = 1 m/s. Then v_B' = 6 - 2(1) = 4 m/s. Ball A continues at 1 m/s, Ball B moves at 4 m/s. Verify KE: 1/2×2×1 + 1/2×1×16 = 1 + 8 = 9 J ✓.
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