BANKED
v₀ = √(rg·tanθ) — the speed at which the road itself steers.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A curve, a bank, a car, and a speed. Highway engineers tilt roads for a reason, and the reason is arithmetic: at the right speed, the road itself does the steering. At any other speed, the tires must make up the difference — up to a point.
The bank
Tilt the road and the normal force tilts with it — and its inward component is a free centripetal force, no friction required. Set the demand equal to that supply and out drops the design speed: v₀ = √(rg tanθ), the speed at which this bank would work on glare ice.
Solve
Away from v₀, friction fills the gap — pointing down-slope when the car is fast and hungry for more turning, up-slope when it is slow and gravity wants to pull it down the bank. Demand against supply, solved both ways, yields a grip window with two exact edges.
Part of a lap
Part of a lap in compressed time: the top view shows the circle being turned; the cross-section shows who is paying — the leaning normal doing the geometry’s share, friction covering the rest. Outside the window, the car drifts off its line (shown presentationally; the verdict is where this physics ends).
Audit
Audited: friction exactly zero at the design speed, demand equal to supply at both window edges to machine precision, and — the engineer’s favorite — mass cancels from every verdict. A truck and a bicycle share the same window; banks are designed for a speed, never a weight.