How to calculate an elastic collision between two bodies: in a 1D collision the final velocities are v1' = ((m1-m2)·v1 + 2·m2·v2)/(m1+m2) and v2' = ((m2-m1)·v2 + 2·m1·v1)/(m1+m2). An elastic collision conserves both momentum p = mv (universal mechanical principle) and kinetic energy K = ½·m·v². Coefficient of restitution e=1 elastic, e=0 perfectly inelastic, 0<e<1 partial.

Formula and physical principles

An elastic collision is an interaction in which total momentum and total kinetic energy are conserved. The 1D final-velocity formulas come from solving simultaneously: m1·v1 + m2·v2 = m1·v1' + m2·v2' (momentum) and ½·m1·v1² + ½·m2·v2² = ½·m1·v1'² + ½·m2·v2'² (kinetic energy). The center-of-mass velocity v_cm = (m1·v1 + m2·v2)/(m1+m2) remains constant. In the COM frame velocities reverse in magnitude (elastic → e=1).

History and scientific context

The laws of collisions were systematically studied by John Wallis, Christopher Wren, and Christiaan Huygens in 1668-1669 at the Royal Society's request. Huygens in De motu corporum ex percussione formulated conservation of mv²/2 for elastic collisions. Newton in Principia (1687) introduced the coefficient of restitution. Conservation of momentum is a general theorem derived from Newton's third law and translation symmetry (Noether's theorem). In relativistic mechanics one conserves total energy-momentum (including m·c²); in quantum mechanics collisions use Born and Schrödinger scattering theory.

Collision types

• Elastic (e=1): K conserved, idealized. Billiard balls, steel spheres, gas atoms.
• Perfectly inelastic (e=0): bodies stick together, K minimum compatible with p conservation. Bullet in ballistic pendulum, cars that lock together.
• Partially inelastic (0<e<1): K partially dissipated. Reality of all macroscopic collisions. Basketballs e=0.75-0.85; tennis e≈0.73-0.78; soccer ball e≈0.52-0.65.
• Super-elastic (e>1): K increases (impossible for purely mechanical, possible in chemical or explosive collisions).
• Limit cases: m1=m2 elastic → velocity swap (cue ball); m1≫m2 → m1 nearly unaffected, m2 rebounds at 2v1.

Worked numerical examples

Example 1 — billiard balls (m1=m2). Cue ball m1=170g at 5 m/s hits black ball m2=170g at rest. Elastic: v1' = 0; v2' = 5 m/s. Cue stops, black launches at same speed. Classic billiards: complete velocity swap between equal masses in head-on elastic collision.

Example 2 — ballistic pendulum (perfectly inelastic). Bullet m1=10g at v1=300 m/s hits block m2=2 kg at rest. Inelastic: v_final = (0.01·300)/(0.01+2.01) = 1.492 m/s. K_initial = 450 J, K_final = 2.24 J. 99.5% K dissipated as heat and deformation. Mass ratio 200× → velocity reduces by same factor.

Example 3 — car crash (partial inelastic). Car m1=1500 kg at 30 m/s rear-ends car m2=1200 kg at rest. Typical sheet-metal e=0.2. v_cm = 16.67 m/s. v1' = 14 m/s; v2' = 20 m/s. 360 kJ dissipated in deformation and heat.

Example 4 — bouncing ball. Basketball m=0.6 kg falls from h=2 m. v_impact = √(2gh) = 6.26 m/s. Ground (m2≈∞) at rest. For e=0.8 (FIBA): v_final = 5.01 m/s upward. Bounce height h' = v²/(2g) = 1.28 m = e²·h. After n bounces: h_n = e^(2n)·h_0.

Example 5 — particle physics 1D. Proton m_p≈938 MeV/c² at 100 MeV on electron m_e≈0.511 MeV/c² at rest. Non-relativistic limit (v_p<<c): exchange nearly nil, electron launches at ~2v_p. In relativistic limit use 4-vectors instead. Classical formula valid for v << c (error <1% if v<0.1c).

Model limits and relativity

1. Classical formulas hold only for v << c (~0.1·c, relativistic error <1%).
2. For relativistic collisions use energy-momentum 4-vectors, conserve (E_tot, p_tot).
3. Coefficient of restitution e depends on materials, speed, temperature.
4. For 2D/3D collisions use vector components and angles; 1D formula insufficient.
5. For rotating rigid bodies angular momentum is also conserved.
6. Explosive/chemical collisions internal energy can increase K (super-elastic).
7. Elementary particles (scattering) need quantum mechanics with cross-section σ.
8. Model assumes instantaneous contact: prolonged collisions need dynamic forces.

Practical applications

1. Sports and billiards: ball trajectories, kicking, tennis, golf.
2. Automotive safety: crash tests (NHTSA, IIHS, Euro NCAP) and crumple zone design.
3. Forensic ballistics: accident reconstruction, projectile trajectories.
4. Particle physics: LHC collider, scattering, new particle detection.
5. Kinetic theory of gases: pressure and temperature from molecular collisions.
6. Robotics: motion planning with contact, object manipulation.
7. Industry: grinding, sieving, mechanical dosing.
8. Astronomy: asteroid collisions, planet formation, gravity assists.
9. Construction: dynamic impact tests on structures, glass impact certification.
10. Civil defense: rockfall, avalanches, collapses (Eurocode EN 1991-1-7 accidental actions).

Common errors

1. Confusing elastic (K conserved) with perfectly inelastic (K minimum).
2. Forgetting velocity signs (vector, not scalar).
3. Applying 1D formulas to 2D/3D without vector decomposition.
4. Confusing restitution coefficient with friction coefficient.
5. Ignoring rotation (spin) for sports balls.
6. Confusing elastic with non-dissipative.
7. Applying classical formulas in relativistic regime.
8. Neglecting heat generation in real collisions (always present for e<1).

Support

Questions: info@calcoi.com. Educational tool for 1D classical collisions. For specialist analysis (crash tests, ballistics, particle scattering) consult a domain expert.

Updated July 2026