Elastic Collisions
Launch balls of different masses and watch momentum and kinetic energy redistribute through perfectly elastic—or perfectly inelastic—collisions. The same physics that governs billiard balls governs particle accelerators.
Click & drag on the canvas to launch a ball — drag length and direction set the velocity. Use the Launch Mass slider to choose ball mass before launching. Right-click a ball to remove it. Toggle Vectors to show momentum arrows, Trails to show paths, and Center of Mass for the system CoM marker. Adjust Restitution from 1.0 (elastic) to 0.0 (perfectly inelastic / stick together).
The physics of collisions
1. Conservation Laws
Two quantities are always conserved in any collision where no external forces act:
momentum and total energy. Momentum, p = mv,
is a vector—it has direction. The total momentum of a closed system never changes,
no matter how violently the objects collide. This is Newton’s third law in disguise:
every action has an equal and opposite reaction, so momentum gained by one object is lost by another.
Kinetic energy, KE = ½mv², is a scalar. In an elastic collision,
kinetic energy is also conserved—no energy goes into deformation, heat, or sound.
Billiard balls and atomic collisions are approximately elastic. In an inelastic collision,
some kinetic energy is converted to other forms. In the extreme case—a perfectly inelastic
collision—the objects stick together and kinetic energy loss is maximized (while momentum
is still conserved).
2. Deriving Post-Collision Velocities
For two objects colliding in 1D with masses m&sub1; and m&sub2;,
initial velocities u&sub1; and u&sub2;, and coefficient of
restitution e, the post-collision velocities are:
v1 = (m1*u1 + m2*u2 + m2*e*(u2 - u1)) / (m1 + m2) v2 = (m1*u1 + m2*u2 + m1*e*(u1 - u2)) / (m1 + m2)
When e = 1 (perfectly elastic), these simplify to the classic formulas.
In 2D, we decompose the collision into components along the line connecting the centers
(the normal direction) and perpendicular to it (the tangential direction). Only the
normal components change; the tangential components are preserved. This is why billiard
balls scatter at angles rather than simply bouncing back.
3. Equal Mass Magic
When two objects of equal mass undergo a head-on elastic collision, something remarkable happens: they exchange velocities completely. A moving ball hitting a stationary ball of the same mass will stop dead, transferring all its motion to the target. This is Newton’s cradle in a nutshell.
In 2D, equal-mass elastic collisions produce another elegant result: the post-collision velocity vectors are always perpendicular—the balls scatter at exactly 90° to each other. This geometric constraint comes directly from the simultaneous conservation of momentum (a vector equation) and kinetic energy (a scalar equation). Try the “Equal Mass” preset to see it.
4. Inelastic Collisions and the Coefficient of Restitution
The coefficient of restitution e measures how “bouncy” a collision is.
It’s defined as the ratio of relative separation speed to relative approach speed along
the line of impact. When e = 1, the collision is perfectly elastic.
When e = 0, the objects stick together. Real collisions fall in between:
a tennis ball on concrete has e ≈ 0.75, a lump of clay has e ≈ 0.
Momentum is conserved regardless of e. Watch the total momentum display as you
lower the restitution—it stays constant while kinetic energy drops. The “lost”
kinetic energy goes into deformation, heat, and sound in the real world.
5. From Billiards to Particle Physics
The same conservation laws that govern billiard balls govern particle collisions at the
Large Hadron Collider. When protons collide at nearly the speed of light, physicists
use conservation of momentum and energy (now relativistic: E² = (pc)² + (mc²)²)
to reconstruct what happened. The Higgs boson was discovered not by seeing it directly,
but by measuring the momenta of its decay products and working backward through
conservation laws.
At macroscopic scales, perfectly elastic collisions are rare—some energy always dissipates. But at the atomic and subatomic level, elastic collisions are the norm. The kinetic theory of gases assumes elastic collisions between molecules, and from this single assumption the Maxwell-Boltzmann speed distribution emerges. Try the “Random Gas” preset and watch the speed distribution evolve toward equilibrium.
6. Center of Mass Frame
Every collision looks simplest from the center-of-mass (CoM) frame. In this frame, the total momentum is zero by definition, so the objects approach each other and recede with equal and opposite momenta. Toggle the CoM indicator to see the system’s center of mass—it moves at constant velocity regardless of how many collisions occur, because no external forces act on the system.
This is a deep insight: the center of mass of a closed system is oblivious to its internal dynamics. It moves in a straight line forever, even as the individual objects ricochet chaotically. The internal motion is separate from the bulk motion—a decomposition that simplifies everything from rocket science to nuclear physics.