What is a banked curve?

A banked curve is a road or track tilted inwards on a bend so that the surface itself helps turn the vehicle. At the ideal angle, the tilt provides all the centripetal force needed, so no friction is required to round the curve at the design speed.

How do you calculate the banking angle?

The ideal banking angle is the inverse tangent of the speed squared divided by the radius times gravity. For 25 metres per second on a 100 metre radius, the angle is the inverse tangent of about 0.637, which is about 32.5 degrees.

Why are race tracks and roads banked?

Banking lets vehicles take corners faster and more safely by using the tilt of the surface, rather than relying only on tyre friction, which can fail when wet or icy. Race tracks, motorway ramps and railway curves are all banked for this reason.

Banked Curve Calculator

A banked curve is a bend in a road or track that is tilted inwards so the surface itself helps turn a vehicle, and this calculator works out the ideal angle of that tilt from the speed and the radius of the turn. Enter the design speed and the radius of the curve, and it returns the banking angle at which a vehicle can round the bend with no help at all from friction, updating as you type. The physics is elegant. To travel in a circle, a vehicle needs a centripetal force pointing towards the centre of the turn, and on a flat road that force comes entirely from the sideways grip of the tyres, which can run out, especially when the road is wet or icy. Banking the surface tilts the support force the road provides so that a component of it points inward, supplying the centripetal force directly. At exactly the right angle, that component is all you need, so the turn can be taken at the design speed even on a frictionless surface, which is why the ideal angle depends only on the speed, the radius and gravity, not on the mass of the vehicle. The formula is the inverse tangent of the speed squared divided by the radius times gravity, so faster speeds and tighter turns call for steeper banking. This is why motor-racing circuits, velodromes, motorway on-ramps and railway curves are all banked, letting them be taken faster and more safely. That makes the tool genuinely useful for physics students learning circular motion, centripetal force and banking and checking homework, and for anyone curious about track and road design. It uses g of 9.81 metres per second squared and assumes the ideal, friction-free angle. The formula and a worked example are explained clearly below.

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Speed (m/s)

Radius of curve (m)

0°

ideal banking angle

Frictionless ideal angle, using g = 9.81 m/s². Real curves add friction for a margin of safety.

How it works

For circular motion the required centripetal acceleration is the speed squared divided by the radius. On a banked surface with no friction, the tangent of the ideal angle equals that acceleration divided by gravity, so the angle is the inverse tangent of the speed squared over the radius times g. The mass of the vehicle does not appear.

Worked example

For a speed of 25 metres per second on a 100 metre radius curve: the speed squared is 625, divided by 100 times 9.81 gives about 0.637. The inverse tangent of 0.637 is about 32.5 degrees, the ideal banking angle.

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