Hong Life CalculatorsUnderstanding the banked curve calculator — concept and principle
A banked curve is a bend whose surface is tilted toward the inside, like a velodrome or a motor-racing oval. Tilting lets part of the road's normal force point toward the centre, so the car can turn without relying entirely on tyre friction. This calculator gives the ideal bank angle at which a chosen speed needs no friction at all, and, for comparison, the maximum speed on a flat (unbanked) curve with a given friction coefficient.
How it is calculated
The speed is entered in km/h and converted to m/s (V = v ÷ 3.6); r is the curve radius in metres and g = 9.80665 m/s².
- Ideal bank angle: θ = arctan(V² ÷ (r g)), in degrees.
- Maximum speed on a flat road: v_max = √(μ g r), converted back to km/h by multiplying by 3.6.
The principle
On a frictionless banked surface, only gravity mg and the normal force N act. The vertical part of N, N cos θ, balances the weight, and the horizontal part, N sin θ, provides the centripetal force mV² ÷ r. Dividing one equation by the other cancels both m and N, leaving tan θ = V² ÷ (rg). At this design speed a car could round the bend even on ice. On a flat road, all the centripetal force must come from friction, which is at most μmg; setting μmg = mv² ÷ r gives v_max = √(μgr). Mass cancels in both results.
Worked example
- Inputs: v = 100 km/h, r = 300 m, μ = 0.7.
- Convert: V = 100 ÷ 3.6 ≈ 27.778 m/s, so V² ≈ 771.60 m²/s².
- tan θ = 771.60 ÷ (300 × 9.80665) ≈ 0.26227, so θ ≈ 14.70°.
- Flat road: √(0.7 × 9.80665 × 300) = √2,059.40 ≈ 45.381 m/s, × 3.6 ≈ 163.37 km/h.
| Speed (km/h) | Ideal bank angle (°) |
|---|---|
| 25 | 0.9391107767 |
| 50 | 3.751409681 |
| 75 | 8.392218846 |
| 100 | 14.69611881 |
| 125 | 22.28387117 |
| 150 | 30.54543058 |
| 200 | 46.37239167 |
How to read the result
14.7° is a slope of about 26%, which is typical of a racing oval, not a public road. Public highways usually keep superelevation to a few percent, often capped around 8–12%, and rely on a combination of bank and a small, comfortable amount of side friction (the design relation e + f = v² ÷ (gR) used in highway guidance). For contrast, oval tracks such as Daytona bank their turns at about 31°. The flat-road limit assumes μ = 0.7, roughly a dry road; on a wet road with μ about 0.4 the same curve's limit falls to about 123 km/h, and on ice far lower.
Common mistakes
- Entering m/s in the km/h field.
- Using the curve's diameter instead of its radius.
- Treating v_max on a flat road as a safe speed; it is the point where tyres start to slide.
- Assuming the ideal angle works at every speed; slower cars tend to slide down and faster cars up.
Frequently asked questions
What happens if I drive slower than the design speed?
The bank provides more inward force than needed, so friction must act up the slope to stop the car sliding down.
What is the maximum speed on a banked curve with friction?
v_max = √(r g (sin θ + μ cos θ) ÷ (cos θ − μ sin θ)), which is larger than both the ideal speed and the flat-road limit.
How do I convert superelevation in percent to degrees?
θ = arctan(e ÷ 100). An 8% superelevation is about 4.57°.
Sources: AASHTO, A Policy on Geometric Design of Highways and Streets (Green Book) · Halliday, Resnick and Walker, Fundamentals of Physics, Wiley