Calculate final velocity in perfectly inelastic collisions using conservation of momentum principle for collision physics analysis.
Classical Mechanics • Collision Physics • 2024
Final Velocity (m/s)
3.333
Total Momentum (kg·m/s)
50.00
Conservation of momentum states: total momentum before collision equals total momentum after collision (in closed isolated system with no external forces). Formula: m₁v₁ + m₂v₂ = (m₁+m₂)v_f for perfectly inelastic collision where objects stick together. Momentum p = mv (mass × velocity, units kg·m/s). Perfectly inelastic: objects combine post-collision (v_final same for both). Elastic: kinetic energy conserved; objects bounce apart. Real collisions: usually between elastic and inelastic (car crash: some deformation, some separation). Collision types: Head-on (opposite velocities), rear-end (same direction), angular (different directions). Historical discovery: Newton's 2nd Law F=ma derived from momentum conservation; principle fundamental to classical mechanics. Applications: vehicle collision investigation (calculate pre-crash velocities from post-crash debris), explosion analysis (rocket propulsion uses momentum conservation—expelled gas momentum equals vehicle momentum change), sports physics (collision between ball-bat, player-player), atomic physics (scattering experiments measure particle masses via momentum transfer). Energy consideration: perfectly inelastic collision loses kinetic energy (converted to deformation, heat, sound); elastic collision conserves kinetic energy. Energy loss in inelastic: ΔE = ½m₁m₂/(m₁+m₂) × (v₁-v₂)². Larger mass difference → greater energy loss in collision.
Advanced applications: center of mass velocity remains constant regardless of collision type. V_cm = (m₁v₁ + m₂v₂)/(m₁+m₂) = constant. In center-of-mass frame, particles approach and recede symmetrically. Recoil dynamics: gun firing bullet—momentum conservation means gun recoils backward. M_gun × v_gun = -m_bullet × v_bullet (opposite directions). Two-body decay (particle physics): daughter particles fly apart conserving parent's momentum. Three-body decay requires vector momentum conservation (Dalitz analysis). Industrial applications: crash test dummies use momentum conservation principles to estimate injury risk. Accident reconstruction: investigators measure skid marks, vehicle deformation, debris distribution to work backward to collision velocities using momentum conservation. Aircraft collision analysis uses same principles. Rocketry: Tsiolkovsky equation F×Δt = Δm×v_exhaust derived from momentum conservation, enabling space travel. Submarine torpedo launch: torpedo momentum change equals momentum transferred to water (Newton's 3rd Law + momentum conservation).
Record Object 1 Properties: Mass m₁ (kg) and velocity v₁ (m/s, positive direction).
Record Object 2 Properties: Mass m₂ (kg) and velocity v₂ (m/s, negative if opposite direction).
Calculate Initial Momentum: p_initial = m₁v₁ + m₂v₂ (vector sum, accounting for directions).
Calculate Total Mass: M_total = m₁ + m₂ (for perfectly inelastic collision).
Apply Conservation Formula: v_final = p_initial / M_total = (m₁v₁ + m₂v₂)/(m₁+m₂). Final velocity in original direction frame.
Scenario: Car (1000 kg @ 5 m/s) collides head-on with truck (2000 kg @ 0 m/s, stationary). Calculate combined velocity post-collision (perfectly inelastic).
Interpretation: Despite truck's higher mass, combined system still moves forward (car's momentum > 0). If truck were moving 2 m/s backward (v₂ = -2 m/s), then p_total = 5000 + 2000(-2) = 1000 kg·m/s → v_final = 0.333 m/s forward. If speeds reversed (car 0 m/s, truck 5 m/s), final velocity 3.33 m/s—lower momentum transfer due to truck's greater mass. Energy dissipated as heat, deformation, sound.
Perfectly inelastic: objects stick (v_final same for both). Elastic: objects bounce (kinetic energy conserved). Real collisions are partially inelastic—some bounce, some deformation.
Yes! Momentum is a vector. Opposite directions require negative velocities. Head-on collision (opposite speeds) produces smaller final momentum than same-direction collision.
Final velocity approaches heavier object's original velocity. 1kg @ 10 m/s hitting 1000kg at rest → final ~0.01 m/s (heavy object barely moves).
Yes! If heavier object moving faster in opposite direction, combined system may reverse direction. Example: truck 2kg @ -10 m/s + car 1kg @ +5 m/s = -1.67 m/s backward.
In closed isolated systems yes. External forces (friction, air resistance, gravity) violate conservation. Short collision duration minimizes external force effects.
Explosion: stationary object splits; products move opposite directions. Momentum still conserved (sum = 0), but kinetic energy increases (chemical energy released).
Crumple zones extend collision time, reducing force. Δp = FΔt; longer Δt = lower F for same Δp. Reduces occupant injuries.
Not directly—assumes discrete objects. Fluid dynamics requires different approach. This applies to solid-body collisions only.
Momentum conservation calculations are essential for collision physics, accident reconstruction, forensic engineering, vehicle safety design, and classical mechanics applications.
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