Keeping the angle of banking, if the radius of curvature is made four times, the percentage increase in the maximum speed with which a vehicle can travel on a circular road is

Options

A. `25%`
B. `50%`
C. `75%`
D. `100%`

Explanation

To solve the problem, we need to understand the relationship between the radius of curvature, the angle of banking, and the maximum speed of a vehicle traveling on a circular road.

### Step-by-Step Solution:

1. **Understanding the Formula for Maximum Speed**:
The maximum speed \( V \) for a vehicle on a banked circular road is given by the formula:
\[ V = \sqrt{R \cdot g \cdot \tan(\theta)} \]
where:
- \( R \) is the radius of curvature,
- \( g \) is the acceleration due to gravity,
- \( \theta \) is the angle of banking.

2. **Initial Speed Calculation**:
Let the initial radius of curvature be \( R \) and the initial maximum speed be \( V_1 \):
\[ V_1 = \sqrt{R \cdot g \cdot \tan(\theta)} \]

3. **New Radius of Curvature**:
If the radius of curvature is made four times, the new radius \( R' \) is:
\[ R' = 4R \]

4. **New Speed Calculation**:
The new maximum speed \( V_2 \) with the new radius of curvature is:
\[ V_2 = \sqrt{R' \cdot g \cdot \tan(\theta)} = \sqrt{4R \cdot g \cdot \tan(\theta)} = \sqrt{4} \cdot \sqrt{R \cdot g \cdot \tan(\theta)} = 2V_1 \]

5. **Calculating the Increase in Speed**:
The increase in speed from \( V_1 \) to \( V_2 \) is:
\[ V_2 - V_1 = 2V_1 - V_1 = V_1 \]

6. **Percentage Increase in Speed**:
The percentage increase in speed is calculated as:
\[ \text{Percentage Increase} = \left( \frac{V_2 - V_1}{V_1} \right) \times 100 = \left( \frac{V_1}{V_1} \right) \times 100 = 100\% \]

### Conclusion:
The percentage increase in the maximum speed with which a vehicle can travel on a circular road, when the radius of curvature is made four times while keeping the angle of banking constant, is **100%**.