How to use it
- Choose whether to find the design speed or the bank angle.
- Enter the radius of the curve and either the bank angle or the speed.
- Enter the friction coefficient between tyres and road: lower for a wet or icy road, 0 to leave friction out.
- Read the design speed or angle, and the speed range in which the car does not slide.
Formulas
- Design speed: v = √(r × g × tan θ), g = 9.80665 m/s²
- Bank angle for a speed: θ = arctan(v² ÷ (r × g))
- Maximum speed with friction μ: v_max² = r × g × (tan θ + μ) ÷ (1 − μ × tan θ)
- Minimum speed: v_min² = r × g × (tan θ − μ) ÷ (1 + μ × tan θ), zero when tan θ ≤ μ
- Sideways acceleration at the design speed: v² ÷ r = g × tan θ
Worked example
A curve of radius 100 m is banked at 15°. tan 15° = 0.26795, so v = √(100 × 9.80665 × 0.26795) = 16.21 m/s = 58.36 km/h. With μ = 0.2 the car stays on line between 28.63 km/h and 79.27 km/h.
For 80 km/h (22.22 m/s) on the same radius the road needs tan θ = 22.22² ÷ (100 × 9.80665) = 0.5036, so θ = 26.73°.
Tips
- Use the radius of the path, not the road width. Large highway curves have radii of hundreds of metres.
- Wet or icy roads shrink the range towards the design speed. On ice only the design speed is safe.
- Racing tracks are banked steeply so that the design speed is high; at low speed cars would slide down without friction.