Momentum Study Guide

📖 Core Concepts Momentum (p) – product of mass and velocity: p = m v (vector). Impulse (J) – change in momentum produced by a force over time: J = Δp = ∫F dt. Conservation of Momentum – total momentum of an isolated system (no external forces) is constant. Reference‑frame dependence – momentum values change with the observer’s inertial frame, but the conservation law holds in every inertial frame. Relativistic momentum – at high speeds: p = γ m₀ v, with γ = 1/√(1 − v²/c²). Four‑momentum – combines energy and 3‑momentum: \(P = \left(\frac{E}{c},\,px,\,py,\,pz\right)\); invariant magnitude \( |P| = m0 c\). --- 📌 Must Remember Units: kg·m/s ( = N·s). Force–momentum relation: F = dp/dt. System momentum: pₜₒₜ = ∑ mᵢ vᵢ = M v₍cm₎. Perfectly inelastic collision velocity: \[ v{\text{final}} = \frac{m1 v{1i}+m2 v{2i}}{m1+m2} \] Coefficient of restitution: \(e = \frac{|v{2f}-v{1f}|}{|v{1i}-v{2i}|}\). Relativistic energy–momentum: \(E^{2} = (pc)^{2} + (m{0}c^{2})^{2}\). Massless particles: \(E = pc\). --- 🔄 Key Processes Impulse calculation Compute \(\displaystyle J = \int{t1}^{t2} \mathbf{F}\,dt\). Set \(J = \Delta \mathbf{p}\) to find final velocity. One‑dimensional elastic collision Apply momentum conservation: \(m1 v{1i}+m2 v{2i}=m1 v{1f}+m2 v{2f}\). Apply kinetic‑energy conservation: \(\frac12 m1 v{1i}^{2}+\frac12 m2 v{2i}^{2}= \frac12 m1 v{1f}^{2}+ \frac12 m2 v{2f}^{2}\). Solve the two equations for \(v{1f}, v{2f}\). Choosing a convenient reference frame Shift to a frame where one particle is initially at rest (lab → CM or particle‑rest). Perform calculations, then transform results back using Galilean (or Lorentz) transformation. Rocket (variable‑mass) momentum balance Write \( \mathbf{F}{\text{ext}} = m\frac{d\mathbf{v}}{dt} - u\frac{dm}{dt}\) (with \(u\) the exhaust speed relative to the rocket). Relativistic momentum update Update γ as velocity changes: \(\gamma = 1/\sqrt{1-v^{2}/c^{2}}\). Compute \(\mathbf{p}= \gamma m0 \mathbf{v}\). --- 🔍 Key Comparisons Elastic vs. Inelastic Collision Elastic: both momentum and kinetic energy conserved. Inelastic: momentum conserved, kinetic energy not conserved (converted to heat, deformation, etc.). Classical vs. Relativistic Momentum Classical: \(\mathbf{p}=m\mathbf{v}\) (valid when \(v \ll c\)). Relativistic: \(\mathbf{p}= \gamma m0 \mathbf{v}\); γ → 1 as \(v \ll c\). Lab Frame vs. Centre‑of‑Mass Frame Lab: velocities are as observed; often messy algebra. CM: total momentum zero; simplifies symmetric collisions. Variable‑mass (rocket) vs. Fixed‑mass Fixed‑mass: \(\mathbf{F}=m\mathbf{a}\). Variable‑mass: extra term \(-u\,dm/dt\) appears. --- ⚠️ Common Misunderstandings “Momentum is conserved even with external forces.” – False; only external‑force‑free (closed) systems conserve momentum. “Impulse equals force times time always.” – Only when the force is constant; otherwise use the integral form. “Relativistic momentum reduces to \(m0 v\) for any speed.” – It reduces to that only when \(v \ll c\); otherwise γ ≠ 1. “Coefficient of restitution > 1 means a “super‑elastic” collision.” – Physically impossible; \(0 \le e \le 1\). --- 🧠 Mental Models / Intuition Momentum as “motion inertia” – Just as mass resists changes in speed, momentum resists changes in direction as well because it’s a vector. Impulse as “push” – Picture a brief shove; the larger the shove (force × time), the bigger the change in the object’s “motion inertia.” CM frame = “balance point” – In the CM frame the system’s total motion is zero; collisions look like two objects bouncing off a stationary point. γ factor as “time‑dilation amplifier” – As speed approaches \(c\), γ blows up, inflating momentum dramatically. --- 🚩 Exceptions & Edge Cases Perfectly inelastic collision – Bodies stick together; kinetic energy loss is maximal (but never negative). Massless particles – No rest mass; momentum still defined via \(p = E/c\). Non‑inertial frames – Momentum conservation does not hold without adding fictitious forces. Relativistic collisions – Must use energy–momentum conservation, not just kinetic energy. --- 📍 When to Use Which Use impulse equation when a known force acts over a short time interval (e.g., a bat hitting a ball). Use conservation of momentum alone for any collision/explosion where external forces are negligible during the interaction. Add kinetic‑energy conservation only for elastic collisions. Switch to CM frame when algebra becomes messy; especially helpful for two‑body problems. Apply relativistic formulas when speeds exceed 0.1 c or when dealing with particles in high‑energy physics. Use variable‑mass rocket equation when mass loss is significant (rockets, leaking containers). --- 👀 Patterns to Recognize “Same‑mass, head‑on elastic collision” → velocities simply swap. “One object initially at rest in CM frame” → final speeds are symmetric about the CM velocity. “e = 0” → perfectly inelastic (stick together). “e = 1” –> perfectly elastic (both momentum and kinetic energy conserved). “High‑speed → γ ≫ 1” → momentum and kinetic energy grow sharply; energy‑momentum relation dominates. --- 🗂️ Exam Traps Distractor: “Momentum is conserved in any collision regardless of external forces.” – Only true for isolated systems. Distractor: “Impulse equals ΔKE.” – Impulse relates to Δp, not Δ kinetic energy. Distractor: “Use \(p = mv\) for rockets.” – Misses the \(-u\,dm/dt\) term; leads to wrong thrust prediction. Distractor: “A coefficient of restitution larger than 1 is possible for super‑elastic collisions.” – Violates energy conservation; maximum is 1. Distractor: “Relativistic momentum formula works for photons with \(m0\neq 0\).” – Photons are massless; use \(E = pc\). ---