Q.Obtain the expression for maximum speed which a vehicle can safely negotiate a curved road banked at an angle $\theta$ with horizontal.

Banking of Roads – From Intuition to Formula

Imagine riding a bicycle on a flat, curved road. To turn, you lean your body inward. If you don't, you feel like you're being thrown outward. That outward push is real — it's the lack of enough centripetal force. On a flat curve, the only force that can pull you inward is friction between the tyres and the road. If the road is wet or you're going too fast, friction isn't enough, and you skid outward.

Now picture a road that is tilted — higher on the outside of the curve, lower on the inside. That tilt is called banking. When you ride on a banked curve, you don't need to lean as much (or at all) because the road itself does the leaning for you. The normal reaction from the road, which is always perpendicular to the surface, now has a horizontal component that points toward the centre of the curve. That horizontal component can provide the centripetal force needed to turn.

The Physics in One Diagram (in words)

On a banked road of angle $\theta$ (the tilt from horizontal), a vehicle of mass $m$ moving with speed $v$ on a curve of radius $r$ experiences:

• Weight $mg$ vertically downward.
• Normal reaction $N$ perpendicular to the road surface.

Resolve $N$ into two components:

• $N \cos\theta$ — vertical component, balances weight.
• $N \sin\theta$ — horizontal component, points toward the centre of the curve.

For the vehicle to stay on the curve without skidding (and assuming no friction), the horizontal component must provide the required centripetal force:

And vertical equilibrium gives:

Dividing the first equation by the second:

This is the ideal banking angle for a given speed $v$ and radius $r$. If the road is banked at exactly this angle, a vehicle can negotiate the curve safely even on a frictionless surface — no sideways friction needed.

What the Formula Tells You

• For a fixed curve radius $r$, a higher speed $v$ requires a steeper bank angle $\theta$.
• For a fixed speed, a sharper curve (smaller $r$) needs a steeper bank.
• The mass $m$ cancels out — a bicycle and a truck both need the same bank angle for the same speed and radius. …