Compute final velocities after a perfectly elastic collision in one dimension. Visualize mass‑scaled spheres, velocity vectors, and verify conservation laws.
Sphere sizes represent mass (log scale). Arrows indicate velocity direction & magnitude (normalized to canvas). Positive = right, negative = left.
An elastic collision is an encounter between two bodies in which the total kinetic energy of the system is conserved. In classical mechanics, elastic collisions also conserve momentum. This idealization applies remarkably well to atomic particles, billiard balls (approximately), and many Newton's cradle demonstrations. The defining condition: coefficient of restitution e = 1.
Conservation of momentum: m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′
Conservation of kinetic energy: ½ m₁v₁² + ½ m₂v₂² = ½ m₁v₁′² + ½ m₂v₂′²
Final velocities (1D): v₁′ = ( (m₁ − m₂)v₁ + 2 m₂ v₂ ) / (m₁ + m₂) v₂′ = ( (m₂ − m₁)v₂ + 2 m₁ v₁ ) / (m₁ + m₂)
The study of collisions dates back to Newton's Principia (1687), where he formulated the laws of impact. The modern derivation of elastic collision formulas relies on solving simultaneous equations of momentum and kinetic energy. In particle physics, elastic scattering (e.g., Rutherford scattering) is fundamental. In engineering, elastic collision principles guide car crumple zone designs (though real collisions are inelastic) and pendulum impact tests.
Beyond simple spheres, elastic collision theory extends to gas kinetic theory (Maxwell–Boltzmann distributions) and even neutron moderation in nuclear reactors. Our calculator handles the pure 1D case — the foundation for understanding more complex 2D/3D collisions via component decomposition.
The canvas displays two spheres scaled by mass (square‑root scaling to keep radii readable). The green arrows indicate initial velocity vectors: longer arrows correspond to higher speed; direction (right/left) is determined by sign. After calculation, the numerical final velocities are displayed, and you can verify that total momentum and kinetic energy remain identical (within floating‑point precision).
| Scenario | Mass ratio | Initial velocities | Outcome highlight |
|---|---|---|---|
| Equal masses | m₁ = m₂ | v₁ = 3, v₂ = -1 | Velocities exchange: v₁′ = -1, v₂′ = 3 |
| Stationary target (heavy) | m₁ = 1, m₂ = 5 | v₁ = 4, v₂ = 0 | Light ball rebounds backward; heavy moves forward slowly |
| Stationary target (light) | m₁ = 5, m₂ = 1 | v₁ = 2, v₂ = 0 | Heavy continues forward; light shoots ahead faster |
| Head‑on collision | m₁ = 3, m₂ = 2 | v₁ = 5, v₂ = -4 | Both reverse or reduce speed depending on masses |
In nuclear reactors, fast neutrons (mass ≈ 1 u) collide elastically with moderator nuclei (e.g., hydrogen, mass ≈ 1 u, or carbon, mass ≈ 12 u). Using the elastic collision formula, a neutron loses maximum energy when colliding with a nucleus of equal mass (hydrogen). For carbon (mass 12), the neutron retains most of its energy. Our calculator lets engineers simulate moderator efficiency: set m₁ = 1 (neutron), m₂ = 12 (carbon), v₂ = 0. After one collision, the neutron’s velocity fraction is (m₁−m₂)/(m₁+m₂) = −11/13 ≈ 0.846, losing about 28% of kinetic energy per collision.
In two dimensions, the collision geometry adds impact parameter and scattering angles. Momentum is conserved separately in x and y axes, and kinetic energy remains conserved. While this calculator focuses on 1D (collinear) collisions — which apply to head‑on impacts or motion along a line — the principles extend to oblique collisions where velocities are resolved into normal and tangential components. For further exploration, see our companion tools for oblique collisions (coming soon).
Our calculator assumes perfect 1D motion and no external forces (friction, air drag). In laboratory experiments, even near‑elastic collisions (e.g., two steel balls) show tiny energy losses due to sound and internal vibrations, but the theoretical values match measured results within 1–2%.