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

Step 1: Understanding the Question:
The question asks for the expression of the optimum (safe) speed at which a vehicle can travel on a curved banked road without depending on friction.
Step 2: Key Formula or Approach:
On a banked road of angle \(\theta\), the horizontal component of the normal reaction (\(N \sin \theta\)) provides the necessary centripetal force, while the vertical component (\(N \cos \theta\)) balances the weight.
Step 3: Detailed Explanation:
Consider a vehicle of mass \(m\) on a road of radius \(r\) banked at angle \(\theta\):
1. Balancing vertical forces: \(N \cos \theta = mg\)

2. Providing centripetal force: \(N \sin \theta = \frac{mv^2}{r}\)
Dividing equation (2) by equation (1):
\[ \frac{N \sin \theta}{N \cos \theta} = \frac{mv^2/r}{mg} \]
\[ \tan \theta = \frac{v^2}{rg} \]
Rearranging for \(v\):
\[ v^2 = rg \tan \theta \implies v = \sqrt{rg \tan \theta} \]
Step 4: Final Answer:
The safety speed is given by \(v = \sqrt{rg \tan \theta}\).