physicshub.github.io

# PhysicsHub simulation context

> Expanded educational context for the interactive simulations listed in [llms.txt](https://physicshub.github.io/llms.txt). This file is generated from the same catalogue and overview data rendered by the site.

## [Bouncing Ball](https://physicshub.github.io/simulations/BouncingBall)

A ball moves in a straight line at constant speed until it meets a wall, then rebounds. It is the simplest possible model of motion and collision: velocity, reflection, and — once gravity and restitution are switched on — energy loss on each bounce. Watching one velocity component flip at every wall makes the idea of a vector concrete.

- Educational levels: Middle School, Elementary
- Difficulty: Core
- Topics: Kinematics, Collision

### Controls
- Mass
- Ball diameter
- Gravity
- Coefficient of restitution

### Key concepts
- Velocity as a vector
- Uniform (constant-velocity) motion
- Elastic reflection off a surface
- Coefficient of restitution

### Governing formulas
- Uniform motion: `x = x_0 + v\,t` (https://physicshub.github.io/formulas#uniform-motion)
- Bounce off a vertical wall: `v_x \rightarrow -e\,v_x` (https://physicshub.github.io/formulas#coefficient-of-restitution)

## [Vector Operations Calculator](https://physicshub.github.io/simulations/VectorsOperations)

Vectors are the language of physics: force, velocity, acceleration and displacement are all vectors. Drag two vectors in the plane and see their sum, difference and dot product update live, alongside each vector's magnitude and direction. It is a sandbox for building intuition before vectors appear in every later topic.

- Educational levels: High School, Middle School
- Difficulty: Core
- Topics: Math, Vectors, Trigonometry

### Controls
- Vector A magnitude
- Vector A angle
- Operation (add / subtract / dot)
- Scalar multiplier

### Key concepts
- Vector magnitude and direction
- Tip-to-tail addition and subtraction
- Components along x and y
- The dot product and the angle between vectors

### Governing formulas
- Magnitude of a vector: `|\vec{a}| = \sqrt{a_x^{2} + a_y^{2}}` (https://physicshub.github.io/formulas#vector-magnitude)
- Sum of two vectors: `\vec{a} + \vec{b} = (a_x + b_x,\; a_y + b_y)` (https://physicshub.github.io/formulas#vector-sum)
- Dot product: `\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y = |\vec{a}||\vec{b}|\cos\theta` (https://physicshub.github.io/formulas#dot-product)

## [Ball Acceleration](https://physicshub.github.io/simulations/BallAcceleration)

A ball is pulled toward your cursor: the further away the cursor, the longer the ball accelerates and the faster it is moving when it arrives. This is a hands-on picture of Newton's second law — a constant force produces a constant acceleration, and acceleration is the rate at which velocity changes, not the velocity itself.

- Educational levels: Middle School, High School
- Difficulty: Core
- Topics: Kinematics, Acceleration, Interactive

### Controls
- Ball diameter
- Maximum speed
- Acceleration

### Key concepts
- Acceleration as the rate of change of velocity
- Newton's second law, F = ma
- Uniformly accelerated motion
- Why velocity and acceleration can point in different directions

### Governing formulas
- Newton's second law: `\vec{F} = m\,\vec{a}` (https://physicshub.github.io/formulas#newtons-second-law)
- Velocity under constant acceleration: `v = v_0 + a\,t` (https://physicshub.github.io/formulas#velocity-constant-acceleration)
- Displacement under constant acceleration: `x = x_0 + v_0 t + \tfrac{1}{2} a t^{2}` (https://physicshub.github.io/formulas#displacement-constant-acceleration)

## [Ball Gravity](https://physicshub.github.io/simulations/BallGravity)

A ball falls under gravity, bounces off the floor and loses a little energy on every impact. Adjust the gravitational acceleration, the ball's mass, the wind, the friction and the restitution to see how each one changes the motion. It connects free fall, Newton's laws and the coefficient of restitution in a single scene.

- Educational levels: High School, Middle School
- Difficulty: Core
- Topics: Dynamics, Gravity, Collision

### Controls
- Mass
- Ball diameter
- Gravity
- Wind acceleration
- Friction coefficient
- Coefficient of restitution

### Key concepts
- Free fall and gravitational acceleration g
- Weight versus mass
- Coefficient of restitution
- Kinetic and gravitational potential energy

### Governing formulas
- Weight: `W = m\,g` (https://physicshub.github.io/formulas#weight)
- Speed after a free fall: `v = \sqrt{2 g h}` (https://physicshub.github.io/formulas#free-fall-speed)
- Restitution at a bounce: `e = \dfrac{v_{\text{after}}}{v_{\text{before}}}` (https://physicshub.github.io/formulas#coefficient-of-restitution)

## [Spring Mass System Simulator](https://physicshub.github.io/simulations/SpringConnection)

A mass hangs from a spring and oscillates. The spring pulls back with a force proportional to how far it is stretched (Hooke's law), and that linear restoring force is exactly what produces simple harmonic motion. Change the mass, the spring constant or the damping and watch the period and decay respond.

- Educational levels: High School, Middle School
- Difficulty: Extended
- Topics: Dynamics, Springs, Oscillations

### Controls
- Bob mass
- Spring constant
- Damping coefficient
- Rest length
- Gravity

### Key concepts
- Hooke's law
- Simple harmonic motion
- Angular frequency, period and frequency
- Damped oscillation

### Governing formulas
- Hooke's law: `F = -k\,x` (https://physicshub.github.io/formulas#hookes-law)
- Angular frequency of a mass on a spring: `\omega = \sqrt{\dfrac{k}{m}}` (https://physicshub.github.io/formulas#spring-angular-frequency)
- Period of a mass on a spring: `T = 2\pi\sqrt{\dfrac{m}{k}}` (https://physicshub.github.io/formulas#spring-period)

## [Simple Pendulum Simulation](https://physicshub.github.io/simulations/SimplePendulum)

A bob on a string swings back and forth about its lowest point. For small swings the motion is simple harmonic and the period depends only on the string length and gravity — not on the mass, and only weakly on the amplitude, the property Galileo called isochronism. Increase the initial angle to see the small-angle approximation break down.

- Educational levels: High School, Middle School
- Difficulty: Core
- Topics: Dynamics, Oscillations, Energy

### Controls
- Length
- Mass
- Gravity
- Damping
- Initial angle

### Key concepts
- Restoring torque of gravity
- Small-angle approximation, sin θ ≈ θ
- Isochronism for small amplitudes
- Energy conservation between the top and bottom of the swing

### Governing formulas
- Period of a simple pendulum: `T = 2\pi\sqrt{\dfrac{L}{g}}` (https://physicshub.github.io/formulas#pendulum-period)
- Pendulum equation of motion: `\ddot{\theta} + \dfrac{g}{L}\sin\theta = 0` (https://physicshub.github.io/formulas#pendulum-equation-of-motion)
- Pendulum speed at the bottom: `v = \sqrt{2 g L\,(1 - \cos\theta_0)}` (https://physicshub.github.io/formulas#pendulum-speed-bottom)

## [Projectile & Parabolic Motion](https://physicshub.github.io/simulations/ParabolicMotion)

Launch a projectile and watch its path. With gravity as the only force, horizontal and vertical motion are independent: the horizontal velocity stays constant while the vertical velocity changes at g, and the combination traces a parabola. Add drag or wind to see how the ideal trajectory deforms.

- Educational levels: High School
- Difficulty: Core
- Topics: Kinematics, Gravity, Math

### Controls
- Launch speed
- Launch angle
- Start height
- Gravity
- Quadratic drag
- Wind acceleration

### Key concepts
- Independence of horizontal and vertical motion
- Parabolic trajectory
- Range, maximum height and time of flight
- Effect of launch angle (45° for maximum range in a vacuum)

### Governing formulas
- Projectile trajectory: `y = x\tan\theta - \dfrac{g\,x^{2}}{2\,v_0^{2}\cos^{2}\theta}` (https://physicshub.github.io/formulas#projectile-trajectory)
- Range of a projectile: `R = \dfrac{v_0^{2}\sin 2\theta}{g}` (https://physicshub.github.io/formulas#projectile-range)
- Maximum height of a projectile: `H = \dfrac{v_0^{2}\sin^{2}\theta}{2g}` (https://physicshub.github.io/formulas#projectile-max-height)

## [Inclined Plane](https://physicshub.github.io/simulations/InclinedPlane)

A block rests on a ramp. Gravity still points straight down, but the surface splits it into a component along the slope that drives sliding and a component into the slope that sets the normal force — and therefore the maximum friction. Change the angle to find the point where static friction can no longer hold the block.

- Educational levels: High School
- Difficulty: Core
- Topics: Dynamics, Forces, Friction

### Controls
- Mass
- Plane angle
- Static friction μₛ
- Kinetic friction μₖ
- Applied force
- Gravity

### Key concepts
- Resolving weight into components
- Normal force as a constraint
- Static versus kinetic friction
- The angle of repose, tan θ = μₛ

### Governing formulas
- Weight along a slope: `F_\parallel = m g \sin\theta` (https://physicshub.github.io/formulas#incline-parallel-force)
- Normal force on a slope: `N = m g \cos\theta` (https://physicshub.github.io/formulas#incline-normal-force)
- Maximum static friction: `f_{s,\max} = \mu_s N = \mu_s m g \cos\theta` (https://physicshub.github.io/formulas#static-friction-max)

## [Circular Motion](https://physicshub.github.io/simulations/CircularMotion)

An object moves around a circle at constant speed. Its velocity is constant in size but always changing direction, so it is always accelerating — toward the centre. That centripetal acceleration needs a real inward force; remove it and the object flies off along the tangent, not outward.

- Educational levels: High School, University
- Difficulty: Extended
- Topics: Dynamics, Kinematics, Vectors

### Controls
- Radius
- Tangential speed
- Mass
- Ball diameter

### Key concepts
- Uniform circular motion
- Centripetal acceleration points to the centre
- Centripetal force comes from a real force (tension, gravity, friction)
- Angular velocity and period

### Governing formulas
- Centripetal acceleration: `a_c = \dfrac{v^{2}}{r} = \omega^{2} r` (https://physicshub.github.io/formulas#centripetal-acceleration)
- Centripetal force: `F_c = \dfrac{m v^{2}}{r}` (https://physicshub.github.io/formulas#centripetal-force)
- Period of circular motion: `T = \dfrac{2\pi r}{v} = \dfrac{2\pi}{\omega}` (https://physicshub.github.io/formulas#circular-period)

## [Three Body Problem](https://physicshub.github.io/simulations/ThreeBody)

Three masses attract each other through Newtonian gravity. Unlike the two-body problem, this system has no general closed-form solution: the motion is chaotic, and tiny changes in the starting positions or velocities lead to completely different futures. The simulation integrates the equations numerically with small sub-steps to stay accurate through close approaches.

- Educational levels: University, High School
- Difficulty: Advanced
- Topics: Gravity, Kinematics, Forces

### Controls
- Starting configuration
- Body mass
- Gravitational constant
- Body size
- Randomness

### Key concepts
- Newton's law of universal gravitation
- Superposition of forces from multiple bodies
- Sensitivity to initial conditions (deterministic chaos)
- Numerical integration and conservation checks

### Governing formulas
- Gravitational force between two bodies: `\vec{F}_{ij} = -\,G\,\dfrac{m_i m_j}{|\vec{r}_{ij}|^{2}}\,\hat{r}_{ij}` (https://physicshub.github.io/formulas#newtons-law-of-gravitation)
- Net gravitational force: `\vec{F}_i = \sum_{j \ne i} \vec{F}_{ij}` (https://physicshub.github.io/formulas#gravitational-superposition)

## [Horizontal Spring](https://physicshub.github.io/simulations/HorizontalSpring)

A mass on a frictionless horizontal surface is tied to a wall by a spring. With gravity removed from the direction of motion, the spring force alone governs the oscillation — the cleanest possible demonstration of Hooke's law and simple harmonic motion. Drag the mass to set an amplitude and release it.

- Educational levels: High School, Middle School
- Difficulty: Core
- Topics: Dynamics, Springs, Oscillations

### Controls
- Bob mass
- Spring constant
- Damping coefficient
- Rest length

### Key concepts
- Hooke's law on a horizontal axis
- Simple harmonic motion
- Amplitude, period and phase
- Total mechanical energy stays constant without friction

### Governing formulas
- Hooke's law: `F = -k\,x` (https://physicshub.github.io/formulas#hookes-law)
- Simple harmonic motion: `x(t) = A\cos(\omega t + \varphi)` (https://physicshub.github.io/formulas#shm-position)
- Total energy: `E = \tfrac{1}{2} k A^{2}` (https://physicshub.github.io/formulas#elastic-potential-energy)

## [Double Pendulum](https://physicshub.github.io/simulations/DoublePendulum)

A second pendulum hangs from the end of the first. The system has just two degrees of freedom, yet its motion is famously chaotic: run it twice from almost-identical starts and the arms soon diverge. This simulation integrates the exact Lagrangian equations for the two angles rather than using a constraint solver, so the chaos is physical and not a numerical artefact.

- Educational levels: University
- Difficulty: Advanced
- Topics: Dynamics, Oscillations, Energy

### Controls
- Length 1 and 2
- Mass 1 and 2
- Gravity
- Damping
- Initial angles

### Key concepts
- Coupled oscillators
- Degrees of freedom and generalized coordinates
- Deterministic chaos and sensitivity to initial conditions
- Energy conservation in the absence of damping

### Governing formulas
- Lagrangian: `\mathcal{L} = T - V` (https://physicshub.github.io/formulas#lagrangian)
- Euler–Lagrange equation: `\dfrac{d}{dt}\dfrac{\partial \mathcal{L}}{\partial \dot{q}_i} - \dfrac{\partial \mathcal{L}}{\partial q_i} = 0` (https://physicshub.github.io/formulas#euler-lagrange)

## [1D Collision simulation](https://physicshub.github.io/simulations/CollisionSimulation)

Two bodies move along a line and collide. Momentum is always conserved in the collision; kinetic energy is conserved only if the collision is perfectly elastic. Set the masses, the initial velocities and the restitution, then read off the momentum and kinetic energy before and after to see which quantities survive.

- Educational levels: High School, Middle School
- Difficulty: Core
- Topics: Collision, Vectors

### Controls
- Ball 1 mass and velocity
- Ball 2 mass and velocity
- Coefficient of restitution

### Key concepts
- Conservation of linear momentum
- Elastic versus inelastic collisions
- Coefficient of restitution
- Centre-of-mass frame

### Governing formulas
- Conservation of momentum: `m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2` (https://physicshub.github.io/formulas#momentum-conservation)
- Elastic collision, final velocity of body 1: `v_1 = \dfrac{m_1 - m_2}{m_1 + m_2}\,u_1 + \dfrac{2 m_2}{m_1 + m_2}\,u_2` (https://physicshub.github.io/formulas#elastic-collision)
- Coefficient of restitution: `e = \dfrac{v_2 - v_1}{u_1 - u_2}` (https://physicshub.github.io/formulas#coefficient-of-restitution)

## [Pi From Collisions](https://physicshub.github.io/simulations/PiCollisions)

Two blocks slide on a frictionless floor with a wall on one side. When the left block is heavier than the right by a factor of 100ᴺ, the total number of collisions equals the first N+1 digits of π. It is a startling link between elastic-collision conservation laws and geometry — the collisions trace out an angle on a circle in velocity space.

- Educational levels: University, High School
- Difficulty: Extended
- Topics: Math, Forces, Dynamics

### Controls
- Mass of small block
- Mass of large block
- Initial velocities
- Block sizes

### Key concepts
- Perfectly elastic collisions in one dimension
- Conservation of momentum and kinetic energy
- Configuration/velocity space and rotations
- Why the count is exact (no collision may be missed)

### Governing formulas
- π from colliding blocks: `N_{\text{collisions}} = \lfloor \pi \cdot 10^{N} \rfloor` (https://physicshub.github.io/formulas#pi-collision-count)
- Conservation of momentum: `m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2` (https://physicshub.github.io/formulas#momentum-conservation)
- Kinetic energy: `K = \tfrac{1}{2} m v^{2}` (https://physicshub.github.io/formulas#kinetic-energy)

## [Trigonometric Circle](https://physicshub.github.io/simulations/TrigonometricCircle)

Drag the angle θ around the unit circle and watch every trigonometric function appear as a real length: sine and cosine as the coordinates of the point, tangent on the vertical line x = 1, and their reciprocals as related segments. A generalized sine wave A·sin(ωθ + φ) plots alongside, tying the circle to the graph.

- Educational levels: Middle School, High School
- Difficulty: Core
- Topics: Math, Trigonometry

### Controls
- Function shown
- Rotation speed
- Initial angle
- Amplitude A
- Frequency ω
- Phase φ

### Key concepts
- The unit circle definition of sine and cosine
- Tangent, cotangent, secant and cosecant as segments
- Amplitude, angular frequency and phase of a sine wave
- Inverse trigonometric functions

### Governing formulas
- Pythagorean identity: `\sin^{2}\theta + \cos^{2}\theta = 1` (https://physicshub.github.io/formulas#pythagorean-identity)
- Tangent as a ratio: `\tan\theta = \dfrac{\sin\theta}{\cos\theta}` (https://physicshub.github.io/formulas#tangent-ratio)
- Sine wave: `y = A\sin(\omega\theta + \varphi)` (https://physicshub.github.io/formulas#sine-wave)

## [Ray Tracing](https://physicshub.github.io/simulations/RayTracing)

A ray tracer makes an image by following rays of light backwards: from the camera, through each pixel, into the scene. At the first surface a ray hits, the laws of geometric optics decide the colour — how squarely the surface faces the light, how far away the light is, whether something casts a shadow, and what the surface reflects. Click any pixel to follow its ray and see every equation with its live numbers.

- Educational levels: University, High School
- Difficulty: Core
- Topics: Optics, Vectors, Interactive

### Controls
- Camera, sphere and light positions (drag them in the diagram)
- Field of view
- Sphere radius
- Light intensity I₀
- Ambient, diffuse and specular coefficients
- Shininess n and reflectivity ρ

### Key concepts
- Rays as the model of geometric optics
- Ray–sphere intersection and the discriminant
- Lambert's cosine law
- The inverse-square law for a point source
- Shadows as blocked straight-line paths
- The law of reflection and specular highlights

### Governing formulas
- Parametric ray: `\vec{P}(t) = \vec{O} + t\,\hat{D}` (https://physicshub.github.io/formulas#ray-equation)
- Ray–sphere intersection: `\Delta = b^2 - 4ac,\quad b = 2\,\hat{D}\cdot(\vec{O}-\vec{C}),\quad c = |\vec{O}-\vec{C}|^2 - r^2` (https://physicshub.github.io/formulas#ray-sphere-intersection)
- Inverse-square law of light: `E = \dfrac{I_0}{d^2}` (https://physicshub.github.io/formulas#inverse-square-law)
- Lambert's cosine law: `I_d = k_d\,E\,\max(0,\ \hat{N}\cdot\hat{L})` (https://physicshub.github.io/formulas#lamberts-cosine-law)
- Law of reflection: `\hat{D}_r = \hat{D} - 2(\hat{D}\cdot\hat{N})\,\hat{N}` (https://physicshub.github.io/formulas#law-of-reflection)

## [Kirchhoff's Circuit Laws](https://physicshub.github.io/simulations/KirchhoffCircuit)

Four DC circuits — a single loop, three resistors in series, three in parallel, and a two-loop network driven by two cells — solved live by Kirchhoff's laws. Charge carriers drift along each wire at a speed set by that branch's current, resistors glow with the power they dissipate, and the readout walks each loop term by term so you can watch the voltages add up to exactly zero.

- Educational levels: High School, Middle School
- Difficulty: Extended
- Topics: Electromagnetism, Energy, Interactive

### Controls
- Circuit layout
- EMF of each cell
- Internal resistance of each cell
- The three resistances

### Key concepts
- Kirchhoff's current law (junction rule) as charge conservation
- Kirchhoff's voltage law (loop rule) as energy conservation
- Ohm's law and equivalent resistance in series and parallel
- EMF, internal resistance and terminal voltage
- Power delivered by a cell and dissipated as heat
- Sign conventions: a negative branch current means a cell is charging

### Governing formulas
- Ohm's law: `V = IR` (https://physicshub.github.io/formulas#ohms-law)
- Junction rule (KCL): `\sum_{\text{into node}} I = \sum_{\text{out of node}} I` (https://physicshub.github.io/formulas#kirchhoffs-current-law)
- Loop rule (KVL): `\sum_{\text{closed loop}} \Delta V = 0` (https://physicshub.github.io/formulas#kirchhoffs-voltage-law)
- Current in a branch of EMF E and resistance R: `I = \dfrac{V_{\text{from}} - V_{\text{to}} + E}{R}` (https://physicshub.github.io/formulas#ohms-law)
- Terminal voltage of a real cell: `V_{\text{term}} = E - I r` (https://physicshub.github.io/formulas#emf-terminal-voltage)
- Power dissipated in a resistor: `P = I^{2} R` (https://physicshub.github.io/formulas#joule-heating)