A national roadway bridge over a canal is in the form of an arc of a circle of radius 49 m. What is the maximum speed with which a car can move without leaving the ground at the highest point? (Take,`g=9.8ms^(-2)`)

Options

A. `19.6ms^(-1)`
B. `40ms^(-1)`
C. `22ms^(-1)`
D. None of the above

Explanation

To solve the problem of determining the maximum speed with which a car can move without leaving the ground at the highest point of a circular arc bridge, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Forces at the Highest Point:**
At the highest point of the arc, the car experiences two forces: the gravitational force acting downwards (mg) and the normal force (N) acting upwards. For the car to just stay in contact with the bridge, the normal force must be zero. This means that the gravitational force provides the necessary centripetal force to keep the car moving in a circular path.

2. **Setting Up the Equation:**
The centripetal force required to keep the car moving in a circular path is given by:
\[ F_c = \frac{mv^2}{r} \]
where:
- \(m\) is the mass of the car,
- \(v\) is the speed of the car,
- \(r\) is the radius of the circular arc (49 m).

At the highest point, the only force providing this centripetal force is the weight of the car (mg). Therefore, we can set up the equation:
\[ mg = \frac{mv^2}{r} \]

3. **Canceling Mass:**
Since the mass \(m\) appears on both sides of the equation, we can cancel it out (assuming \(m \neq 0\)):
\[ g = \frac{v^2}{r} \]

4. **Rearranging the Equation:**
Rearranging the equation to solve for \(v^2\):
\[ v^2 = rg \]

5. **Substituting Values:**
Now, substitute the values for \(r\) and \(g\):
- \(r = 49 \, \text{m}\)
- \(g = 9.8 \, \text{m/s}^2\)

Thus,
\[ v^2 = 49 \times 9.8 \]

6. **Calculating \(v^2\):**
Calculate \(v^2\):
\[ v^2 = 480.2 \]

7. **Taking the Square Root:**
Finally, take the square root to find \(v\):
\[ v = \sqrt{480.2} \approx 21.9 \, \text{m/s} \]

8. **Conclusion:**
Therefore, the maximum speed with which a car can move without leaving the ground at the highest point of the bridge is approximately **22 m/s**.