A road is 10 m wide. Its radius of curvature is 50 m . The outer edge is above the lower edge by a distance of 1.5 m . This road is most suited for the velocity

Options

A. `2.5m//sec`
B. `4.5 m//sec`
C. `6.5 m//sec`
D. `8.5 m//sec`

Answer

To find the most suited velocity for a vehicle on a curved road with a given radius of curvature and height difference, we can follow these steps:

### Step-by-step Solution:

1. **Understand the Geometry of the Road:**
- The road is 10 m wide, with a radius of curvature of 50 m.
- The outer edge of the road is elevated by 1.5 m compared to the inner edge.

2. **Identify the Angle of Inclination (θ):**
- The height difference (h) is 1.5 m and the width of the road (w) is 10 m.
- The angle of inclination can be approximated using the tangent function:
\[ \tan(\theta) = \frac{h}{w} = \frac{1.5}{10} = 0.15 \]

3. **Relate the Forces Acting on the Vehicle:**
- When a vehicle moves along the curved road, the forces acting on it include the gravitational force (mg) and the normal force (N).
- The component of the normal force acting towards the center of the curve provides the necessary centripetal force for circular motion.

4. **Set Up the Equations:**
- The vertical component of the normal force balances the weight of the vehicle:
\[ N \cos(\theta) = mg \]
- The horizontal component of the normal force provides the centripetal force:
\[ N \sin(\theta) = \frac{mv^2}{r} \]
- Here, \(r\) is the radius of curvature (50 m), and \(v\) is the velocity we want to find.

5. **Divide the Two Equations:**
- By dividing the two equations, we eliminate \(N\):
\[ \frac{N \sin(\theta)}{N \cos(\theta)} = \frac{mv^2/r}{mg} \]
- This simplifies to:
\[ \tan(\theta) = \frac{v^2}{rg} \]

6. **Substitute the Values:**
- We know \(g \approx 9.8 \, \text{m/s}^2\), \(r = 50 \, \text{m}\), and \(\tan(\theta) = 0.15\):
\[ 0.15 = \frac{v^2}{50 \times 9.8} \]

7. **Solve for Velocity (v):**
- Rearranging gives:
\[ v^2 = 0.15 \times 50 \times 9.8 \]
- Calculating:
\[ v^2 = 0.15 \times 490 = 73.5 \]
- Taking the square root:
\[ v = \sqrt{73.5} \approx 8.57 \, \text{m/s} \]

8. **Conclusion:**
- The most suited velocity for the vehicle on this curved road is approximately **8.5 m/s**.