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The period of a pendulum is T = 2π√(L/g) — it depends on the string length and gravity, but not on the mass. Try maxing out gravity and watch the swing speed up. Move it to "moon gravity" (1.6 m/s²) and watch it crawl.

With no air resistance, range is maximised at 45°. Notice how 30° and 60° give the same range — every angle has a "complement" that lands in the same spot. Velocity hits range twice as hard as you might think (it scales with v²).

Universal indicator turns red in strong acids, yellow-green near neutral, and violet in strong bases. Each pH unit means a 10× change in hydrogen ion concentration — that's why a small pH shift is a big chemical deal.

Reaction rate climbs with both temperature (faster particles, harder collisions) and concentration (more particles, more chances to collide). The classic Arrhenius relationship is roughly exponential in temperature — rule of thumb: every 10 K doubles the rate.

Amplitude stretches the wave vertically. Frequency squashes it horizontally — higher frequency, more wiggles per unit. Phase slides it left or right. These three numbers describe every sound, light, and ocean wave you'll ever meet.

For any right-angled triangle, a² + b² = c² — the squares on the two short sides add up to the square on the hypotenuse. Drag the corner around: the equation always balances. The orange and green squares (visualised on the legs) always equal a square on side c.

Photosynthesis needs light and CO₂ — and the rate is bottlenecked by whichever is in shorter supply (the limiting factor). Crank one to max while keeping the other low and you'll see diminishing returns. Real plants follow the same rules.

The logistic equation dN/dt = r·N·(1 − N/K) is what populations actually do. Early on it grows exponentially — but as N approaches the carrying capacity K, growth slows and eventually stops. Same shape as a bacterial colony, a viral outbreak, or rabbits on an island.