Banked Curve
A car on a banked curve at adjustable angle and friction; find the ideal speed where no friction is needed.
Published: June 8, 2026 · Updated: June 9, 2026
Objective
Explore how banking a road at angle θ provides centripetal force without friction at a single ideal speed v_ideal = sqrt(r·g·tan θ). Verify that this frictionless cornering speed is independent of the car's mass, depends only on bank angle and curve radius, and that friction is required — with a finite skid limit — whenever the car drives faster or slower than ideal.
Setup
- Set Bank Angle to 20°, Curve Radius to 80 m, Car Speed to 14 m/s, and Friction Coefficient to 0.4. Note the Ideal Speed readout and the direction of the friction arrow on the cross-section diagram.
- Click Start and observe the Friction Force readout. Pause the sim, then drag Car Speed slowly upward toward the Ideal Speed value shown in the HUD — watch the friction arrow shrink and the Friction Force readout approach zero.
- Continue dragging speed past the ideal value. Observe that the friction arrow reverses direction and Friction Force becomes positive (up-slope).
- Reset and set Bank Angle to 0°. Note that v_ideal = 0 and the car requires friction for any nonzero speed — a flat road provides no centripetal component from the normal force.
- Reset, set Bank Angle to 45°, Curve Radius to 200 m, and find the new ideal speed. Compare it to the formula sqrt(200 × 9.81 × tan 45°) ≈ 44.3 m/s.
- Set Friction Coefficient to 0 and observe that the safe-band shading on the graph collapses to a single vertical line at v_ideal — no margin exists without friction.
Analytical Prediction
At Bank Angle = 20°, Curve Radius = 80 m, the frictionless ideal speed is:
At Car Speed = 20 m/s (above ideal), the required friction force with mass = 1200 kg is:
The maximum safe speed with μ = 0.4 is:
Note that mass cancels in every speed-limit formula — v_ideal, v_max, and v_min are all mass-independent.
Results Analysis
After starting the sim, check the Ideal Speed (m/s) readout — it should read approximately 16.9 m/s for the default settings (20°, 80 m). Drag Car Speed to match this value; the Friction Force (N) readout should drop to near 0 and the '✓ No friction' annotation should appear on the cross-section diagram. The secondary graph's live amber dot will sit on the curve at its zero-crossing (the forest-green v_ideal dashed line). Setting speed to 20 m/s should raise Friction Force to roughly 1612 N ± 2 N, consistent with the prediction. The Max Safe Speed readout should show approximately 26.5 m/s. Centripetal Accel at 20 m/s, 80 m radius reads 5.00 m/s² (= 20²/80).
Source of Error
This sim models a point mass on a frictionless banked road cross-section with no aerodynamic drag, no tire compliance, and no longitudinal (braking or throttle) forces. The normal force formula N = mg/cos θ assumes the car is in vertical equilibrium, which excludes suspension dynamics. The friction force formula treats the tire–road contact as a rigid Coulomb interface with a single coefficient μ — real tires have load-dependent, speed-dependent, and combined-slip grip curves. No rolling resistance is included. These idealizations mean v_ideal and the skid limits are exact within the model; the residual between the HUD readouts and the worked-example predictions is therefore purely numerical, not physical.
Further Exploration
- Set μ = 0 and sweep Bank Angle from 0° to 60°. At what angle does v_ideal first reach highway speed (≈ 28 m/s) for r = 80 m? Does the safe speed band collapse to a single point?
- With Bank Angle = 30° and Curve Radius = 200 m, compute v_ideal analytically (≈ 33.7 m/s) — can you reach it with the speed slider? What happens to v_max and v_min as μ increases from 0 to 0.8?
- Set Bank Angle to 60° and Curve Radius to 300 m. The formula predicts v_ideal ≈ 71 m/s — above the slider ceiling. How does the graph's zero-crossing move off-chart, and what does the v_ideal marker do?
- Is the ideal speed affected when you change the friction coefficient from 0 to 0.8? Why does v_ideal not depend on μ even though friction is present at other speeds?
- Compare two runs archived as ghosts: first with θ = 20°, r = 80 m; then with θ = 20°, r = 160 m. Does the zero-crossing shift by a factor of sqrt(2) as the formula predicts?