1D Frontal Collisions

Simulate elastic and inelastic collisions between two particles with coefficient of restitution

Input parameters

Particle 1 mass

Mass of the left particle

0.15000

Particle 2 mass

Mass of the right particle

0.15000

Initial velocity 1

Velocity of particle 1 (positive → right)

-300300

Initial velocity 2

Velocity of particle 2 (positive → right)

-300300

Collision type

Coefficient of restitution

e = 1 elastic, e = 0 perfectly inelastic

Energy unit

Collision visualization

e = 1

Before impactt = 0.00 sAnimation time

Ek₁ 100 JEk₂ 22.5 J

Speed

Position vs time

x₁(t)x₂(t)x_cm(t)

Kinetic energy balanceConserved 100.0% / Dissipated 0.0%

Per-particle distribution (live)P1 81.6% / P2 18.4%

Results

Case analysis

Elastic collision

With e = 1, both momentum and kinetic energy are conserved. Results: v₁' = -8.57 m/s and v₂' = 4.43 m/s.

Final velocity 1

-8.571

m/s

Final velocity 2

4.429

m/s

Center of mass velocity

0.7143

m/s

Initial kinetic energy

122.5 J

Final kinetic energy

122.5 J

Dissipated energy

Total linear momentum

5 kg·m/s

Momentum conservation

P = 5 kg·m/s ✓

Fundamentals & Explanation

Conservation Laws and Fundamental Symmetries

The kinematics of collisions are governed by the principles of conservation of linear momentum and energy. According to Noether's theorem (1915), the conservation of linear momentum is a direct mathematical consequence of the homogeneity of space (spatial translation symmetry). In the absence of net external forces, the total momentum of the system must remain invariant over time, regardless of the nature of the internal interaction forces during the impact.

Thermodynamic Clarification: Newton's coefficient of restitution (e) is a phenomenological macroscopic approximation. At the microscopic (atomic) level, all fundamental collisions are perfectly elastic. The macroscopic "loss" of kinetic energy (when e < 1) represents the dispersion of that energy into unobservable internal degrees of freedom: heat (phonons), lattice vibrations, or irreversible plastic deformation.

Mathematical Formulation and Coefficient of Restitution

Sir Isaac Newton experimentally postulated in 1687 that the relative velocity of separation of two spheres is proportional to their relative velocity of approach. This kinematic relationship, coupled with momentum conservation, forms a linear system of equations that uniquely determines the final velocities.

Linear Momentum (P)

m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}

Restitution (e)

e = -\frac{v_{1f} - v_{2f}}{v_{1i} - v_{2i}}

Solving the system yields the closed-form expressions implemented in this simulator:
v₁f = [m₁v₁i + m₂v₂i + m₂e(v₂i - v₁i)] / (m₁ + m₂)

Galilean Invariance and the Center of Mass (CM) Frame

The laws of classical mechanics are invariant under Galilean transformations. Analyzing the collision from the Center of Mass reference frame (which moves at a constant velocity V_cm = P / M_total) drastically simplifies the problem: in the CM frame, the total momentum is exactly zero at all times.

In the CM system, particles always collide head-on with opposite momenta and rebound maintaining opposite momenta. If the collision is elastic (e = 1), they simply reverse their velocity vector in this reference frame.

Physical Limits of the Model

Relativistic Limit (v → c)

When velocities approach the speed of light (c), linear momentum ceases to be p = m·v and requires the Lorentz factor (γ). The apparent inertial mass increases and Galilean velocity addition must be replaced by Lorentz transformations. The simulator outputs warnings if v > 0.1c.

Quantum Limit

At subatomic scales, the concept of deterministic trajectories collapses due to the uncertainty principle. Collisions are described using scattering theory, where particles are treated as wave packets and the probabilistic "differential cross-section" is calculated.

Mass Limit Cases

m₂ ≫ m₁ (Wall): Mass m₂ barely changes its velocity. m₁ bounces back with almost the same approach velocity.

m₁ = m₂, e=1: Perfect exchange of momentum and energy. Particle 1 stops and transfers all its inertia to particle 2.

Dissipation (ΔE_k)

The dissipated energy is maximized when e = 0, but 100% of E_k is never lost if the total momentum (P) is not zero, due to the inertia of the Center of Mass.

Academic References

[1] Goldstein, H., Poole, C., & Safko, J. (2001). Classical Mechanics (3rd ed.). Addison Wesley. Sections 3.11 and 3.12 (Kinematics of Scattering).
[2] Landau, L. D., & Lifshitz, E. M. (1976). Mechanics: Course of Theoretical Physics, Vol. 1 (3rd ed.). Butterworth-Heinemann.
[3] Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica. "Axioms, or Laws of Motion", Corollary VI (Conservation of Momentum).
[4] Noether, E. (1918). Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 235-257. (Symmetry-Conservation Link).

1D Frontal Collisions

Input parameters

Collision visualization

Results

Elastic collision

Fundamentals & Explanation

Conservation Laws and Fundamental Symmetries

Mathematical Formulation and Coefficient of Restitution

Galilean Invariance and the Center of Mass (CM) Frame

Physical Limits of the Model

Relativistic Limit (v → c)

Quantum Limit

Mass Limit Cases

Dissipation (ΔE_k)

Academic References

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Conservation Laws and Fundamental Symmetries

Mathematical Formulation and Coefficient of Restitution

Galilean Invariance and the Center of Mass (CM) Frame

Physical Limits of the Model

Relativistic Limit (v → c)

Quantum Limit

Mass Limit Cases

Dissipation (ΔE_k)

Academic References