Question

What is the formula for the safety speed on a banked road?

• A. \(v = \sqrt{rg\sin\theta}\)
• B. \(v = \sqrt{\frac{rg}{\tan\theta}}\)
• C. \(v = \sqrt{rg\tan\theta}\)
• D. \(v = rg\tan\theta\)

Hint:

For a frictionless banked road, the safe velocity is \(v = \sqrt{rg\tan\theta}\). This speed prevents the vehicle from skidding either upward or downward.

Answer 1

The Correct Option is C

Concept: When a vehicle moves along a curved banked road, the road is inclined at an angle \(\theta\). This inclination helps provide the necessary centripetal force required for circular motion without relying on friction. For a vehicle moving safely without slipping, the required centripetal force is provided by the horizontal component of the normal reaction.

Step 1: Forces acting on the vehicle. Two main forces act on the vehicle:
• Weight \(mg\) acting vertically downward
• Normal reaction \(N\) from the road surface Resolving the normal reaction: \[ N\cos\theta = mg \] \[ N\sin\theta = \frac{mv^2}{r} \]

Step 2: Dividing the equations. \[ \frac{N\sin\theta}{N\cos\theta} = \frac{mv^2/r}{mg} \] \[ \tan\theta = \frac{v^2}{rg} \]

Step 3: Solving for velocity. \[ v = \sqrt{rg\tan\theta} \] Thus, the safety speed on a banked road is given by \[ v = \sqrt{rg\tan\theta} \]