# Collision Simulator - Elastic & Inelastic, Restitution, 2D Glancing

> Simulate 1D cart collisions with any masses, velocities and restitution e, see momentum and KE before and after, plus 2D billiard collisions.

- URL: https://qilingtools.com/en/math/collision
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

In 1D mode two carts collide on a track. Set their masses, their velocities and the coefficient of restitution e (1 is perfectly elastic, 0 means they stick together) and watch the collision play out. A table lists each cart's velocity, momentum and kinetic energy before and after, along with the energy lost, the centre-of-mass velocity and the impulse. The 2D mode is a glancing billiard collision. The impact parameter b sets how thin the hit is, and you see both balls' outgoing speeds and angles and the vector triangle in which the final momenta add up to the initial one. Both modes can switch to the centre-of-mass frame, where the physics is clearest.

## How to use

1. Choose the 1D cart collision or the 2D billiard collision, optionally starting from a preset
2. Set the masses, velocities (or impact parameter b in 2D) and the coefficient of restitution e
3. Press Play to watch, scrub the timeline, and tick “Centre-of-mass frame” to change viewpoint
4. Read the momentum and kinetic-energy table, the energy lost and the outgoing angles

## Features

- Restitution e from 0 to 1 covers elastic, inelastic and perfectly inelastic collisions
- Before and after table of velocity, momentum and KE, with energy lost and impulse
- KE split into centre-of-mass and relative parts, showing how much can be lost
- 2D glancing collisions with impact parameter, deflection and opening angles
- Momentum vector triangle, with the elastic-collision circle for p₁′
- Switch between the lab frame and the centre-of-mass frame

## FAQ

### Why is momentum always conserved but kinetic energy not?

During a collision the two objects push on each other with equal and opposite forces. As long as external forces (or their impulse) are negligible, total momentum is conserved. But objects deform, heat up and make noise, turning some kinetic energy into other forms, so KE usually decreases. Only a perfectly elastic collision (e = 1) conserves KE. A perfectly inelastic one (e = 0), where the objects move off together, loses the most.

### How is the coefficient of restitution defined, and how much energy is lost?

The coefficient of restitution is the ratio of separation speed after to approach speed before: e = −(v₂′ − v₁′)/(v₂ − v₁). The energy lost is ΔKE = ½μ(v₁ − v₂)²(1 − e²), where μ = m₁m₂/(m₁+m₂) is the reduced mass. Total KE splits into the energy of the centre-of-mass motion, ½Mv_cm², and the energy of the relative motion, ½μv_rel². Momentum conservation locks in the first part, so no collision can remove it. A collision can only take away the fraction 1 − e² of the second.

### Why do equal masses swap velocities in an elastic head-on collision?

Put m₁ = m₂ and e = 1 into the formulas and you get v₁′ = v₂ and v₂′ = v₁: the velocities are exchanged. This is how Newton's cradle works. The moving ball stops dead and the ball it hits leaves with its speed. When a heavy object hits a light one, both move forward and the light one goes faster. When a light object hits a heavier one at rest, the light one bounces back.

### Why do billiard balls separate at 90°?

For equal masses, an elastic collision and a target at rest, momentum conservation gives p = p₁′ + p₂′ and energy conservation gives p² = p₁′² + p₂′². Both can hold only if p₁′ is perpendicular to p₂′, so the balls leave at 90° (in a head-on hit the cue ball simply stops). In the momentum diagram, the tip of p₁′ always lies on the circle with p₁ as its diameter. With unequal masses or energy loss, the angle is no longer 90°.

### What is the impact parameter b?

It is the perpendicular distance from the incoming ball's line of motion to the centre of the target ball. b = 0 is a head-on hit, and the larger b, the thinner the hit, until the balls miss when b ≥ r₁ + r₂. For smooth balls the impulse acts along the line of centres at contact. That line makes an angle φ with the incoming direction where sin φ = b/(r₁+r₂), and the struck ball leaves along it. That's why the aiming point in pool decides where the object ball goes.

## Related tools

- [Unit Converter](https://qilingtools.com/en/math/unit): Free online unit converter: length, area, volume, weight, temperature, time, speed, data, pressure, energy and angle — one value converts every unit at once.
- [Vector Calculator](https://qilingtools.com/en/math/vector): Free online vector calculator for 2D and 3D: addition, scalar multiples, dot and cross products, magnitude, angle, unit vector and projections, with checks.
- [Significant Figures Calculator](https://qilingtools.com/en/math/significant-figures): Free online sig fig calculator: count significant figures digit by digit, round to N sig figs or decimals with six rounding modes, and compute by sig fig rules.
- [Tone Generator](https://qilingtools.com/en/audio/tone-generator): Free online tone generator: sine, square, sawtooth and triangle waves from 1 to 20,000 Hz, a note picker, sweeps, white/pink/brown noise and a WAV download.
- [Audio Spectrogram](https://qilingtools.com/en/audio/spectrogram): Free online spectrogram generator: see time, frequency and level from a Hann-windowed STFT, with FFT size, log axis, dB range and colour maps. Exports PNG.
- [Decibel Meter](https://qilingtools.com/en/audio/decibel): Free online decibel meter: measure noise with your mic in real time, A/C/Z weighting, fast/slow response, min/max/average/Leq, 60-second graph and calibration.
- [Physics Formula Calculator](https://qilingtools.com/en/math/physics): Physics formulas solved with steps: SUVAT, projectiles, Newton’s laws, inclines, energy, Ohm’s law, resistors, RC, buoyancy, waves, lenses, gravity.
- [Fourier Series & FFT Visualizer](https://qilingtools.com/en/math/fourier): Fourier series coefficients and animated partial sums for square, sawtooth, triangle or any function (Gibbs), draw-your-own epicycles and FFT spectra.