The radius of the curved road on a national highway is `R`. The width of the road is `b`. The outer edge of the road is raised by `h` with respect to the inner edge so that a car with velocity `v` can pass safe over it. The value of `h` is

Options

A. `(v^(2)b)/(Rg)`
B. `(v)/(Rgb)`
C. `(v^(2)R)/(g)`
D. `(v^(2)b)/(R )`

Explanation

To find the height \( h \) of the outer edge of the curved road with respect to the inner edge, we can use the principles of circular motion and the geometry of the road.

### Step-by-Step Solution:

1. **Understanding the Geometry:**
- The road is curved with a radius \( R \).
- The width of the road is \( b \).
- The outer edge is raised by a height \( h \) compared to the inner edge.
- The angle of the banked road is \( \theta \).

2. **Relating Height and Width:**
- The angle \( \theta \) can be expressed using the tangent function:
\[ \tan \theta = \frac{h}{b} \]
- This means:
\[ h = b \tan \theta \]

3. **Forces Acting on the Car:**
- When a car is moving on the banked road, it experiences two forces: gravitational force \( mg \) acting downwards and the normal force \( N \) acting perpendicular to the surface of the road.
- There is also a centripetal force required for the car to move in a circular path, which is provided by the horizontal component of the normal force.

4. **Setting Up the Equations:**
- The components of forces can be resolved as follows:
- The vertical component:
\[ N \cos \theta = mg \]
- The horizontal component (centripetal force):
\[ N \sin \theta = \frac{mv^2}{R} \]

5. **Dividing the Equations:**
- From the two equations, we can eliminate \( N \):
\[ \frac{N \sin \theta}{N \cos \theta} = \frac{\frac{mv^2}{R}}{mg} \]
- This simplifies to:
\[ \tan \theta = \frac{v^2}{gR} \]

6. **Relating \( h \) and \( b \):**
- Substituting the expression for \( \tan \theta \) into the equation for \( h \):
\[ \tan \theta = \frac{h}{b} = \frac{v^2}{gR} \]
- Therefore, we can express \( h \) in terms of \( b \):
\[ h = b \cdot \frac{v^2}{gR} \]

7. **Final Expression for \( h \):**
- Thus, the height \( h \) can be calculated as:
\[ h = \frac{v^2 b}{gR} \]

### Conclusion:
The value of \( h \) is given by:
\[ h = \frac{v^2 b}{gR} \]