Banking of Roads — Ideal Speed

Why roads are banked

On a flat curved road, only friction provides the centripetal force. Friction can fail (wet road, worn tires), causing the vehicle to skid.

A banked road tilts inward at the curve. Now the normal force from the road has a horizontal component pointing toward the centre of the curve. This inward component can provide centripetal force — without any friction.

The angle $\theta$ at which the road is banked, for a given speed $v$ and radius $r$:

$\tan\theta = \frac{v^2}{rg}$

At this "ideal speed", no friction is needed at all.

Derivation

Consider a vehicle of mass $m$ on a banked road of banking angle $\theta$. No friction acts (ideal condition).

Forces on the vehicle:

• Weight $mg$ downward
• Normal force $N$ perpendicular to the banked surface (tilted inward from vertical)

Resolve $N$:

• Vertical component: $N\cos\theta$ (balances weight)
• Horizontal component: $N\sin\theta$ (provides centripetal force)

Vertical equilibrium:

$N\cos\theta = mg \quad \text{...(1)}$

Horizontal — centripetal:

$N\sin\theta = \frac{mv^2}{r} \quad \text{...(2)}$

Divide (2) by (1):

$\tan\theta = \frac{v^2}{rg}$

$\boxed{\theta = \tan^{-1}\left(\frac{v^2}{rg}\right)}$

The ideal speed for a given bank angle

$v_{ideal} = \sqrt{rg\tan\theta}$

At this speed, friction does zero work and experiences zero wear — the road and vehicle are perfectly matched.

Design of roads and racetracks

Road engineers use this formula to decide the banking angle for a given design speed and curve radius.

Example: A highway curve of radius $r = 200$ m, design speed $v = 20$ m/s ($72$ km/h):

$\tan\theta = \frac{20^2}{200 \times 10} = \frac{400}{2000} = 0.2$

$\theta = \tan^{-1}(0.2) \approx 11.3°$

Racing circuits (like Formula 1) use much steeper banking to allow very high speeds through corners.

What happens at non-ideal speeds

At $v < v_{ideal}$: the vehicle tends to slide down the bank (inward). Friction acts up the bank (outward) to prevent this.

At $v > v_{ideal}$: the vehicle tends to slide up the bank (outward). Friction acts down the bank (inward) to prevent this.

At exactly $v_{ideal}$: no tendency to slide either way — zero friction.

The range of safe speeds (with friction) is derived in the "Banking with Friction" entry.