Select the correct equation for optimum speed of a car on a banked road of angle of banking `theta` and radius of curvature r.

Options

A. `v= sqrt(rg)`
B. `v=sqrt(rg sin theta)`
C. `v= sqrt(rg tan theta)`
D. `v= sqrt(rg cos theta)`

Answer

To determine the optimum speed of a car on a banked road, we can derive the equation step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a banked road with an angle of banking \( \theta \) and a radius of curvature \( r \).
- The objective is to find the optimum speed \( V_0 \) at which a car can travel on this banked road without relying on friction.

2. **Forces Acting on the Car**:
- When a car is moving on a banked road, the forces acting on it include gravitational force \( mg \) (acting downwards) and the normal force \( N \) (acting perpendicular to the surface of the road).
- The car experiences centripetal acceleration due to its circular motion.

3. **Components of Forces**:
- The normal force can be resolved into two components:
- \( N \cos \theta \) acts vertically (balancing the weight).
- \( N \sin \theta \) acts horizontally (providing the necessary centripetal force).

4. **Applying Newton's Second Law**:
- Vertically: The vertical component of the normal force balances the weight of the car:
\[ N \cos \theta = mg \]
- Horizontally: The horizontal component provides the centripetal force required for circular motion:
\[ N \sin \theta = \frac{mv^2}{r} \]

5. **Eliminating Normal Force**:
- From the first equation, we can express \( N \):
\[ N = \frac{mg}{\cos \theta} \]
- Substituting this expression for \( N \) into the second equation:
\[ \frac{mg}{\cos \theta} \sin \theta = \frac{mv^2}{r} \]
- Simplifying this gives:
\[ g \tan \theta = \frac{v^2}{r} \]

6. **Solving for Optimum Speed**:
- Rearranging the equation to solve for \( v \):
\[ v^2 = rg \tan \theta \]
- Taking the square root of both sides gives:
\[ v = \sqrt{rg \tan \theta} \]

7. **Final Expression**:
- Thus, the optimum speed \( V_0 \) of the car on a banked road is given by:
\[ V_0 = \sqrt{rg \tan \theta} \]

### Conclusion:
The correct equation for the optimum speed of a car on a banked road is:
\[ V_0 = \sqrt{rg \tan \theta} \]