Calculate elastic and inelastic collision outcomes. Find final velocities, kinetic energy loss, and momentum conservation for two-body collisions and explosions.
The law of conservation of momentum states that the total momentum of an isolated system remains constant before and after a collision: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'. This fundamental principle, rooted in Newton's third law, applies to every collision — from subatomic particles to galaxy mergers.
This calculator handles three collision types: elastic (kinetic energy conserved), inelastic (kinetic energy partially lost), and explosions (objects separate from rest). For inelastic collisions, you can set the coefficient of restitution (0 = perfectly inelastic, 1 = elastic) to model real-world impacts. The calculator provides final velocities, momentum verification, and complete energy analysis.
Presets include pool balls, car crashes, bullet-into-block problems, Newton's cradle, head-on collisions, and grenade explosions. Before-vs-after summary tables and mass-ratio effect tables deepen the analysis. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results.
Momentum conservation calculations are essential in collision analysis (accident reconstruction, ballistics), physics education, engineering design (crash safety, sports equipment), and computational physics simulations.
This calculator handles all three collision regimes and provides before/after tables, energy accounting, and mass-ratio effects — far more than a simple formula substitution tool. Keep these notes focused on your operational context. Tie the context to the calculator’s intended domain.
p = m₁v₁ + m₂v₂ (conserved). Elastic: v₁' = ((m₁-m₂)v₁ + 2m₂v₂)/(m₁+m₂). Inelastic with restitution e: v₁' = (m₁v₁ + m₂v₂ + m₂e(v₂-v₁))/(m₁+m₂). Perfectly inelastic (e=0): v' = (m₁v₁ + m₂v₂)/(m₁+m₂).
Result: v₁' = 0 m/s, v₂' = 2 m/s
Two equal-mass pool balls: the cue ball stops and the target ball moves away at the cue ball's original speed. This perfect momentum and energy transfer is characteristic of equal-mass elastic collisions.
Elastic collisions preserve total kinetic energy — translational motion is fully transferred between objects. Inelastic collisions convert some kinetic energy to thermal energy, sound, and permanent deformation. The coefficient of restitution (e) quantifies this: e = (v₂' − v₁') / (v₁ − v₂). Real car crashes have e ≈ 0.1–0.3; baseball bat hits have e ≈ 0.5.
Momentum conservation is one of three fundamental conservation laws in classical mechanics, alongside energy conservation and angular momentum conservation. Each corresponds to a symmetry: momentum conservation arises from translational symmetry (physics is the same everywhere in space), per Noether's theorem.
Forensic engineers use momentum conservation to reconstruct vehicle crashes. Pre-crash speeds are calculated from post-crash trajectories and deformation. Combined with skid mark analysis and energy methods (crush energy), momentum analysis provides court-admissible speed estimates with typical accuracy of ±10%.
In elastic collisions, both momentum AND kinetic energy are conserved — objects bounce off perfectly. In inelastic collisions, momentum is conserved but kinetic energy is lost to heat, sound, and deformation. Most real collisions are inelastic.
The coefficient of restitution (e) measures how "bouncy" a collision is: e = 1 for perfectly elastic, e = 0 for perfectly inelastic (objects stick together), and 0 < e < 1 for real collisions. A rubber ball on concrete has e ≈ 0.8; a car crash has e ≈ 0.1-0.3.
Not macroscopically — all real collisions lose some energy. However, collisions between hard billiard balls (e ≈ 0.95), steel ball bearings, and subatomic particles come very close. At the atomic level, gas molecule collisions are effectively elastic.
Total momentum is zero before the collision (equal and opposite momenta). Conservation of momentum requires zero total momentum after — so in a perfectly inelastic collision, both cars stop, and ALL kinetic energy is lost to deformation. This is the most destructive collision type.
A bullet embeds in a block (perfectly inelastic collision). By measuring the block's velocity after impact, you can calculate the bullet's original speed: v_bullet = (m_bullet + m_block) × v_after / m_bullet. The energy "lost" goes into heat and deformation of the block.
Yes, in an isolated system (no external forces). Gravity, friction, and other external forces transfer momentum to/from the system, but during the brief collision event, external forces are negligible compared to impact forces, so momentum is effectively conserved.