Banked Curve Speed Calculator
When a road is banked at angle θ, there is an ideal speed at which a vehicle can navigate the curve without relying on friction at all. At this speed the horizontal component of the normal force provides exactly the centripetal acceleration needed. The formula is v = sqrt(r g tan θ), where r is the curve radius (m), g is gravitational acceleration (m/s²), and θ is the bank angle in degrees. This is fundamental to road design, railway engineering, and motor racing track design. The calculator also shows the ideal speed in km/h for reference.
Banked curve ideal speed formula
v = √(r × g × tan(θ))
Where r is the curve radius (m), g is gravitational acceleration (m/s²), and θ is the bank angle (degrees). This gives the speed at which centripetal force is provided entirely by the normal force, with no friction required.
Understanding banked curves
- The centripetal acceleration v² / r must be provided by horizontal forces. On a banked curve (no friction) this comes from the horizontal component of normal force: N sin(θ) = mv² / r.
- The vertical component of normal force balances gravity: N cos(θ) = mg. Dividing these equations gives v² = rg tan(θ).
- Typical highway superelevation of 8% (4.6°) on a 300-m radius curve gives ideal speed = sqrt(300 x 9.81 x tan(4.6°)) = 17.4 m/s = 62.6 km/h.
- Banking reduces tire wear and allows higher safe speeds, especially in wet or icy conditions where friction is reduced.
- Train tracks are banked by raising the outer rail, a quantity called cant or superelevation.
Banked curve: frequently asked questions
What is a banked curve?
A banked curve is a road or track section where the outer edge is raised relative to the inner edge. When a vehicle navigates a banked curve at the ideal speed, the normal force from the road provides all the centripetal force needed, eliminating reliance on friction. The formula for this ideal speed is v = sqrt(r g tan theta).
What happens if you drive too fast or too slow on a banked curve?
At the ideal speed (no friction needed), the car stays in its lane through normal force alone. If the speed is too high, friction must act inward to provide extra centripetal force. If too slow, friction must act outward to prevent sliding down the bank. The friction-free formula gives the optimal design speed for the road.
How does banking angle affect the ideal speed?
Higher banking angles allow higher ideal speeds. Since v = sqrt(r g tan theta), and tan theta increases rapidly with angle (tan 45° = 1, tan 60° = 1.73, tan 75° = 3.73), steeply banked tracks (like a velodrome) permit much higher cornering speeds.
Does mass affect the ideal banked curve speed?
No. The mass of the vehicle cancels in the derivation. The ideal speed depends only on the radius of the curve, the bank angle, and gravitational acceleration. This is why all vehicles, regardless of mass, have the same ideal speed on a given banked curve.
What bank angle do highways use?
US highway design standards (AASHTO) typically limit banking (superelevation) to 8-12% (about 4.6° to 6.8°) on regular roads. NASCAR ovals can have banking up to 33° (Bristol Motor Speedway) or even higher.
Official sources
- OpenStax University Physics Volume 1: Centripetal Force.
- NIST Reference on Constants: Standard acceleration of gravity.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.