Why do the balls sometimes just exchange velocities or stop completely?

These are special cases dictated by mass ratios and impact geometry. If the masses are equal and one ball is initially stationary, a head-on collision causes the moving ball to stop and the stationary one to move forward with the original speed. A glancing collision between equal masses scatters them at a 90-degree angle relative to each other. These outcomes are direct mathematical solutions to the conservation laws.

Is this a realistic model for real billiard balls?

It is an excellent first-order model but has simplifications. Real billiard collisions are nearly elastic, but some kinetic energy is lost to sound, heat, and internal vibration. Furthermore, real balls have spin (angular momentum), and friction with the table is crucial for their rolling motion. This simulator strips away these complexities to focus purely on the core momentum and energy principles.

How is the 'angle of collision' determined in the simulation?

The critical angle is not the initial direction of motion, but the angle between the line connecting the centers of the two balls at the moment of impact (the 'line of centers') and the initial velocity vector of the incoming ball. The simulator calculates this geometry internally. Momentum is conserved perpendicular to this line, while the components along the line are exchanged according to the conservation laws.

What does it mean for momentum to be a 'vector' quantity in this context?

A vector has both magnitude and direction. The total momentum of the system is the vector sum of each ball's momentum (mass times velocity vector). Conservation of momentum means this total vector sum is identical before and after the collision. You must account for both the x- and y-components of each ball's velocity separately, which is why changing the collision angle dramatically changes the outcomes.

2D Collisions

Two-dimensional elastic collisions provide a rich context for exploring the fundamental conservation laws of classical mechanics. This simulator models the interaction between two disks, or 'billiard balls,' on a frictionless surface. The core physics is governed by the conservation of momentum and the conservation of kinetic energy. Since the system is isolated from external forces, the total momentum vector remains constant before and after the collision: m₁v⃗₁ᵢ + m₂v⃗₂ᵢ = m₁v⃗₁f + m₂v⃗₂f. For perfectly elastic collisions, kinetic energy is also conserved: ½m₁v₁ᵢ² + ½m₂v₂ᵢ² = ½m₁v₁f² + ½m₂v₂f². By manipulating these equations and resolving velocities into components (typically along the line connecting the centers at impact and perpendicular to it), one can solve for the final velocities. The simulator simplifies reality by assuming perfectly elastic collisions, point-like contact between perfectly rigid bodies, and the absence of friction and rotational motion. This allows students to isolate and master the vector nature of momentum conservation. Interacting with the model, users can adjust initial speeds, masses, and impact angles to observe how these parameters dictate the scattering of the objects. The primary learning outcome is a deep, intuitive understanding of how momentum, a vector quantity, is redistributed in two dimensions while total energy remains unchanged.

Who it's for: High school and introductory college physics students studying momentum, energy, and two-dimensional kinematics, as well as educators seeking a dynamic demonstration tool.

Key terms

Conservation of Momentum
Elastic Collision
Kinetic Energy
Vector Components
Center of Mass Frame
Impulse
Scattering Angle
Coefficient of Restitution

Live graphs

How it works

Two smooth disks on a frictionless rectangular table. Collisions resolve along the normal with coefficient of restitution e; walls are elastic. Momentum is conserved; kinetic energy is conserved only when e = 1 and only ball–ball collisions matter.

Key equations

j = −(1+e)(v₁−v₂)·n̂ / (1/m₁ + 1/m₂), vᵢ′ = vᵢ ± (j/mᵢ)n̂

n̂ points from ball 1 to ball 2 along the line of centers.

Frequently asked questions

Why do the balls sometimes just exchange velocities or stop completely?: These are special cases dictated by mass ratios and impact geometry. If the masses are equal and one ball is initially stationary, a head-on collision causes the moving ball to stop and the stationary one to move forward with the original speed. A glancing collision between equal masses scatters them at a 90-degree angle relative to each other. These outcomes are direct mathematical solutions to the conservation laws.
Is this a realistic model for real billiard balls?: It is an excellent first-order model but has simplifications. Real billiard collisions are nearly elastic, but some kinetic energy is lost to sound, heat, and internal vibration. Furthermore, real balls have spin (angular momentum), and friction with the table is crucial for their rolling motion. This simulator strips away these complexities to focus purely on the core momentum and energy principles.
How is the 'angle of collision' determined in the simulation?: The critical angle is not the initial direction of motion, but the angle between the line connecting the centers of the two balls at the moment of impact (the 'line of centers') and the initial velocity vector of the incoming ball. The simulator calculates this geometry internally. Momentum is conserved perpendicular to this line, while the components along the line are exchanged according to the conservation laws.
What does it mean for momentum to be a 'vector' quantity in this context?: A vector has both magnitude and direction. The total momentum of the system is the vector sum of each ball's momentum (mass times velocity vector). Conservation of momentum means this total vector sum is identical before and after the collision. You must account for both the x- and y-components of each ball's velocity separately, which is why changing the collision angle dramatically changes the outcomes.

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