A car is moving on a horizontal curved road with radius `50m`.The approzimate maximum speed of car will be,if friction between tyres and road is `0.34` [take `g=10ms^(-2)`

Options

A. `3.4ms^(-1)`
B. `13ms^(-1)`
C. `14ms^(-1)`
D. `22.4ms^(-1)]`

Answer

To solve the problem of finding the maximum speed of a car moving on a horizontal curved road, we can use the concept of centripetal force and friction. Here’s the step-by-step solution:

### Step 1: Understand the Forces Involved
When a car is moving on a curved path, the frictional force between the tires and the road provides the necessary centripetal force to keep the car moving in a circle.

### Step 2: Set Up the Equation
The maximum frictional force can be expressed as:
\[ F_{\text{friction}} = \mu \cdot m \cdot g \]
where:
- \( \mu \) is the coefficient of friction (0.34),
- \( m \) is the mass of the car,
- \( g \) is the acceleration due to gravity (10 m/s²).

The centripetal force required to keep the car moving in a circle is given by:
\[ F_{\text{centripetal}} = \frac{m \cdot v^2}{r} \]
where:
- \( v \) is the speed of the car,
- \( r \) is the radius of the curve (50 m).

### Step 3: Set the Forces Equal
For maximum speed, the frictional force must equal the centripetal force:
\[ \mu \cdot m \cdot g = \frac{m \cdot v^2}{r} \]

### Step 4: Cancel Out the Mass
Since mass \( m \) appears on both sides of the equation, we can cancel it out:
\[ \mu \cdot g = \frac{v^2}{r} \]

### Step 5: Solve for Speed \( v \)
Rearranging the equation gives:
\[ v^2 = \mu \cdot g \cdot r \]

### Step 6: Substitute the Values
Now, substitute the known values into the equation:
- \( \mu = 0.34 \)
- \( g = 10 \, \text{m/s}^2 \)
- \( r = 50 \, \text{m} \)

Calculating \( v^2 \):
\[ v^2 = 0.34 \cdot 10 \cdot 50 \]
\[ v^2 = 0.34 \cdot 500 \]
\[ v^2 = 170 \]

### Step 7: Take the Square Root
Now, take the square root to find \( v \):
\[ v = \sqrt{170} \]

### Step 8: Approximate the Square Root
The square root of 170 is approximately:
\[ v \approx 13.04 \, \text{m/s} \]

Since we are looking for an approximate maximum speed, we can round this to:
\[ v \approx 13 \, \text{m/s} \]

### Final Answer
Thus, the approximate maximum speed of the car is:
\[ \boxed{13 \, \text{m/s}} \]