How to use this calculator
Enter the curve radius, the bank angle and the static friction coefficient. To check a particular speed, enter it; add the car’s mass to get the actual forces.
Formulas
Design speed (no friction)v02 = r g tan θ
Speed limits with frictionv2max, min = r g sin θ ± μ cos θcos θ ∓ μ sin θ
Worked example
100 m radius, 15° bank, μs = 0.4 (the default)
- v0 = √(100 × 9.81 × tan 15°) = 16.2 m/s (58 km/h).
- vmax = √(981 × (0.2588 + 0.3864)/(0.9659 − 0.1035)) = 27.1 m/s (97.5 km/h).
- sin 15° < 0.4 cos 15°, so friction can hold a parked car: vmin = 0.
Frequently asked questions
Why bank a curve at all?
On a flat road only friction can turn the car. Tilting the road lets part of the normal force point toward the center, so less friction is needed and the curve stays safe in the wet.
Does the car’s mass matter?
Not for the speeds: mass cancels. It only sets the size of the forces.