Banked Curve Calculator: Banking Angle for a Speed and the Safe Speed Range with Friction
Banking angle and speed limits for a curve.
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How this calculator works
Choose the banking angle or the speed range.
Enter the speed and radius, or the angle and the friction coefficient.
The calculator gives the angle, or the maximum and minimum speeds.
Formulas used
- \tan\theta = \frac{v^2}{r g}
- v_{max}^2 = \frac{r g(\tan\theta + \mu)}{1 - \mu\tan\theta}
- v_{min}^2 = \frac{r g(\tan\theta - \mu)}{1 + \mu\tan\theta}
Worked example
With the default numbers in the calculator above:
- Ideal banking angle (no sideways friction needed at this speed): 32.501 degrees
- Slope of the banking: 0.6371 (63.71 %)
Change the numbers in the calculator to see the full working, every step, for your own case.
A second worked example
A different case, with these numbers: What do you want to find? Maximum and minimum speed for a banking angle and friction.
- Ideal speed with no friction: 18.896 m/s (68.025 km/h)
- Maximum speed before sliding up the bank: 32.189 m/s (115.88 km/h)
- Minimum speed before sliding down the bank: 0 m/s (0 km/h)
Common questions
What is the ideal banking angle?
The angle where the normal force from the road supplies all the force needed to turn the vehicle, so no friction is needed. It depends on the speed and the radius only, not on the mass.
Why does the mass not matter?
The mass cancels out of the force balance, because both the turning force and the weight depend on it.
What if the speed is faster than the design speed?
Friction between the tyres and the road acts down the slope to hold the vehicle on the curve, up to the maximum speed.
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Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any physics or dynamics text (banked curves).
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