For a circular path of radius 300 m, the super elevation is restricted to 0.1 m for a width of 1.6 m. The maximum speed, in m/s, of the vehicle to avoid overturn is _____. (round o... For a circular path of radius 300 m, the super elevation is restricted to 0.1 m for a width of 1.6 m. The maximum speed, in m/s, of the vehicle to avoid overturn is _____. (round off up to 2 decimals)
Understand the Problem
The question is asking to calculate the maximum speed of a vehicle on a circular path while considering super elevation. It requires applying physics concepts related to circular motion and stability to avoid overturning.
Answer
The maximum speed of the vehicle to avoid overturn is approximately $11.07 \, \text{m/s}$.
Answer for screen readers
The maximum speed of the vehicle to avoid overturning is approximately $11.07 , \text{m/s}$.
Steps to Solve
- Identify the Given Values
We have the following information:
- Radius of circular path, $R = 300 , \text{m}$
- Super elevation height, $h = 0.1 , \text{m}$
- Width of the road, $w = 1.6 , \text{m}$
- Calculate the Angle of Super Elevation
The angle of super elevation, $\theta$, can be calculated using the relationship: $$ \tan(\theta) = \frac{h}{w} $$
Substituting the values: $$ \tan(\theta) = \frac{0.1}{1.6} $$
- Determine the Centripetal Acceleration
Using the angle, we can find the centripetal acceleration needed to maintain circular motion: $$ a_c = g \cdot \tan(\theta) $$ where $g \approx 9.81 , \text{m/s}^2$ (acceleration due to gravity).
- Calculate the Maximum Speed
The maximum speed, $v_{max}$, to avoid overturning can be calculated using the formula: $$ v_{max} = \sqrt{g \cdot R \cdot \tan(\theta)} $$
- Substituting Values to Find Final Speed
Substitute the values into the equation to find $v_{max}$:
- Substitute $\tan(\theta)$ from previous step into the equation and calculate $v_{max}$.
The maximum speed of the vehicle to avoid overturning is approximately $11.07 , \text{m/s}$.
More Information
This scenario involves concepts of circular motion and the effects of incline (super elevation) on stability. It demonstrates how physics can be applied to real-world driving scenarios.
Tips
- Miscalculating the Angle: Ensure that you calculate $\tan(\theta)$ correctly using the given super elevation and width.
- Using Incorrect Units: Always check that the units are consistent (e.g., meters for radius and height).
- Forgetting Acceleration Due to Gravity: Always include $g$ when calculating centripetal acceleration.
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