2D Collision

Introduction

A 2D collision is one in which two objects collide at an angle, meaning their motion cannot be described along a single line. Unlike the simpler head-on case, both the horizontal and vertical components of momentum must be tracked and conserved independently. Understanding two-dimensional collisions is essential in fields ranging from billiards and sports science to particle physics and orbital mechanics.

This article focuses on the elastic 2D collision — the idealised case where kinetic energy is also conserved in addition to momentum. The key mathematical technique is decomposing all velocities into x and y components, applying the 1D elastic collision equations along the collision axis, and then recombining the results back into the lab frame.

The Physics Explained

When two objects collide in two dimensions, the governing principle is still conservation of momentum — but now applied as a vector law. Because momentum is a vector, its x-component and y-component are each independently conserved. This means we get two separate conservation equations, one for each axis, rather than just one.

The standard approach is to work in the frame of the collision normal — the line connecting the centres of the two objects at the moment of impact. This line is called the line of action or the contact normal. Velocity components along this axis drive the collision; velocity components perpendicular to it (the tangential components) are completely unaffected in a frictionless elastic collision and pass straight through unchanged.

Along the contact normal, the problem reduces to an ordinary 1D elastic collision. The two normal-direction velocity components are exchanged according to the standard elastic formulas, weighted by the mass ratio. Once those post-collision normal components are calculated, they are vector-added back to the unchanged tangential components to give the final 2D velocity of each object.

The angle θ between the initial velocity of object 1 and the contact normal determines how much energy is transferred. A head-on collision (θ = 0°) produces maximum energy transfer; a glancing blow (θ close to 90°) transfers very little. This is why a pool ball struck off-centre barely changes the target ball's motion.

For an elastic collision between equal masses where one is initially stationary, there is a beautiful geometric result: after the collision, the two velocity vectors are always perpendicular to each other. The angle between the outgoing paths always adds up to exactly 90°. This can be proven using the constraint that both momentum and kinetic energy are conserved simultaneously.

Key Equations

x-momentum conservation m₁·v₁ₓ + m₂·v₂ₓ = m₁·v₁ₓ' + m₂·v₂ₓ'

y-momentum conservation m₁·v₁ᵧ + m₂·v₂ᵧ = m₁·v₁ᵧ' + m₂·v₂ᵧ'

Kinetic energy conservation ½m₁|v₁|² + ½m₂|v₂|² = ½m₁|v₁'|² + ½m₂|v₂'|²

Normal component after collision — object 1 v₁ₙ' = ((m₁ − m₂)·v₁ₙ + 2·m₂·v₂ₙ) / (m₁ + m₂)

Normal component after collision — object 2 v₂ₙ' = ((m₂ − m₁)·v₂ₙ + 2·m₁·v₁ₙ) / (m₁ + m₂)

Tangential components (unchanged) v₁ₜ' = v₁ₜ v₂ₜ' = v₂ₜ

Speed of object 1 (magnitude) |v₁| = sqrt(v₁ₓ² + v₁ᵧ²)

Direction of object 1 after collision θ₁' = arctan(v₁ᵧ' / v₁ₓ')

Key Variables

Symbol	Name	Unit	Meaning
m₁	Mass of object 1	kg	Inertial mass of the first object
m₂	Mass of object 2	kg	Inertial mass of the second object
v₁ₓ, v₁ᵧ	Velocity components of object 1	m/s	Horizontal and vertical velocity of object 1 before collision
v₂ₓ, v₂ᵧ	Velocity components of object 2	m/s	Horizontal and vertical velocity of object 2 before collision
v₁ₙ	Normal velocity component — object 1	m/s	Component of object 1's velocity along the contact normal
v₁ₜ	Tangential velocity component — object 1	m/s	Component of object 1's velocity perpendicular to the contact normal
v₁ₙ', v₂ₙ'	Post-collision normal components	m/s	Normal velocity components after the elastic collision
θ	Collision angle	degrees (°)	Angle between the initial velocity and the contact normal
p	Momentum	kg·m/s	Mass times velocity vector; conserved in all collisions
KE	Kinetic energy	J	½m\|v\|²; conserved only in elastic collisions

Real World Examples

Billiards and pool: When a cue ball strikes a target ball off-centre, the collision is two-dimensional. The angle of deflection depends on the impact parameter — the perpendicular distance between the centres at contact. Equal-mass elastic collisions always produce a 90° angle between the outgoing paths, a fact skilled players use to plan their shots.
Molecular gas collisions: Gas molecules are in constant random motion, colliding with each other at all angles and speeds. The statistical behaviour of these countless 2D and 3D elastic collisions underlies the kinetic theory of gases, explaining temperature, pressure, and diffusion from first principles.
Subatomic scattering experiments: In Rutherford scattering, alpha particles are fired at a gold foil. The angles at which they deflect are governed by 2D elastic collision geometry and the Coulomb force. Rutherford's analysis of these scattering angles led directly to the discovery of the atomic nucleus.
Sports: In snooker, ice hockey, and curling, pucks and balls undergo oblique collisions. Coaches and players intuitively apply 2D collision physics when calculating where a rebounding puck will travel or how to play a cannon shot.

How the Simulation Works

The simulation places two circular objects on a frictionless 2D surface. You can adjust the mass of each object and the initial speed and direction of object 1 using the control sliders. Object 2 starts at rest by default, though you may also give it an initial velocity and angle.

When the two circles overlap, the simulation computes the contact normal — the unit vector pointing from the centre of object 2 to the centre of object 1. Each object's velocity is then decomposed into a normal component (along this axis) and a tangential component (perpendicular to it). The 1D elastic collision formulas are applied to the normal components, and the tangential components are left unchanged. The results are recombined into new 2D velocity vectors for each object.

The simulation displays velocity vectors as arrows on each object so you can watch both direction and magnitude change through the collision. Running totals of total momentum (x and y separately) and total kinetic energy are shown in the readout panel; you will see these quantities remain constant before and after each collision, confirming the conservation laws in action.