Interactive Lab.
Speed as slope, vectors in 3D, and why things mix — three ideas you can poke at, live in the browser.
Three ideas you can poke at. Move a control and watch the picture answer — nothing is a video, every scene is live.
Speed is the slope of distance
A speedometer and a graph tell the same story. Plot how far each car has travelled against time, and the steepness of the line — its slope — is exactly the speed.
Each line on the graph is one car's distance over time. A faster car climbs the graph more steeply — its slope is its speed. Change a slider and watch the car and its line steepen together.
Drag a speed slider: the car and its line steepen together. One fact, two pictures.
Seeing 3D space
A point in space is just three numbers — how far along x, then y, then z. Set them with the sliders and watch the single arrow they build; drag the scene to walk around it.
Three numbers — x, y, z — pin down one arrow in space. The dashed steps walk along each axis in turn to the tip. Drag the scene to orbit; the arrow is the same, you just see it from a new angle.
Orbiting changes your view, never the vector. The three components are the same from every angle.
Diffusion is mixing is entropy
Ink dropped in water spreads and never gathers back. Two colours start apart here; remove the barrier and they mix — because there are vastly more mixed arrangements than separated ones.
Nothing pulls the colors together — each particle just bounces at random. With the barrier gone there are overwhelmingly more mixed arrangements than separated ones, so the mixing meter climbs and never comes back. That is entropy.
The mixing meter only ever rises. That one-way climb is the second law.