A car takes a turn on a slippery road at a safe speed of 9.8 m/s .If the coefficient of friction is 0.2 the manimum radius of the are in which the car takes a trun is

Options

A. 20 m
B. 49 m
C. 24.5 m
D. 80 m

Explanation

To find the minimum radius of the arc in which the car can safely take a turn on a slippery road, we can follow these steps:

### Step 1: Understand the Forces Involved
When a car takes a turn, it experiences a centripetal force that keeps it moving in a circular path. On a slippery road, this centripetal force is provided by the frictional force between the tires and the road.

### Step 2: Write the Equation for Centripetal Force
The centripetal force \( F_c \) required to keep the car moving in a circle is given by the formula:
\[ F_c = \frac{mv^2}{r} \]
where:
- \( m \) is the mass of the car,
- \( v \) is the speed of the car,
- \( r \) is the radius of the turn.

### Step 3: Determine the Frictional Force
The maximum frictional force \( F_f \) that can act on the car is given by:
\[ F_f = \mu N \]
where:
- \( \mu \) is the coefficient of friction,
- \( N \) is the normal force.

On a flat surface, the normal force \( N \) is equal to the weight of the car, \( mg \):
\[ N = mg \]
Thus, the frictional force becomes:
\[ F_f = \mu mg \]

### Step 4: Set the Forces Equal
For the car to safely make the turn, the frictional force must be equal to the centripetal force:
\[ \mu mg = \frac{mv^2}{r} \]

### Step 5: Simplify the Equation
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \mu g = \frac{v^2}{r} \]
Rearranging this gives us:
\[ r = \frac{v^2}{\mu g} \]

### Step 6: Substitute the Known Values
Given:
- \( v = 9.8 \, \text{m/s} \)
- \( \mu = 0.2 \)
- \( g = 9.8 \, \text{m/s}^2 \)

Substituting these values into the equation:
\[ r = \frac{(9.8)^2}{0.2 \times 9.8} \]

### Step 7: Calculate the Radius
Calculating the right-hand side:
\[ r = \frac{96.04}{1.96} = 49 \, \text{m} \]

### Conclusion
The minimum radius of the arc in which the car can safely take a turn is:
\[ \boxed{49 \, \text{m}} \]