A circular road of radius `1000m` has hanging angle `45^(@)` The maximum safe speed (in `ms^(-1)` of a car having a mass `2000kg ` will be (if the coefficient of friction between tying and road is `0.3)`

Options

A. `172`
B. `124`
C. `99`
D. `86`

Explanation

To find the maximum safe speed of a car on a circular road with a given radius and angle of banking, we can use the formula that incorporates both the banking angle and the friction between the tires and the road. Here's a step-by-step solution:

### Step 1: Identify the given values
- Radius of the circular road, \( R = 1000 \, \text{m} \)
- Mass of the car, \( m = 2000 \, \text{kg} \) (not needed for the calculation of speed)
- Angle of banking, \( \theta = 45^\circ \)
- Coefficient of friction, \( \mu = 0.3 \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \) (approximation)

### Step 2: Write the formula for maximum speed on a banked curve with friction
The maximum speed \( V_{\text{max}} \) on a banked road with friction is given by the formula:
\[ V_{\text{max}} = \sqrt{R g \left( \tan \theta + \frac{\mu}{1 - \mu \tan \theta} \right)} \]

### Step 3: Calculate \( \tan \theta \)
Since \( \theta = 45^\circ \):
\[ \tan 45^\circ = 1 \]

### Step 4: Substitute values into the formula
Substituting the known values into the formula:
\[ V_{\text{max}} = \sqrt{1000 \times 10 \left( 1 + \frac{0.3}{1 - 0.3 \times 1} \right)} \]

### Step 5: Simplify the expression inside the square root
Calculate the denominator:
\[ 1 - 0.3 \times 1 = 1 - 0.3 = 0.7 \]
Now substitute this back into the equation:
\[ V_{\text{max}} = \sqrt{1000 \times 10 \left( 1 + \frac{0.3}{0.7} \right)} \]

### Step 6: Calculate \( \frac{0.3}{0.7} \)
\[ \frac{0.3}{0.7} \approx 0.4286 \]
So now we have:
\[ V_{\text{max}} = \sqrt{1000 \times 10 \left( 1 + 0.4286 \right)} = \sqrt{1000 \times 10 \times 1.4286} \]

### Step 7: Calculate the product
\[ 1000 \times 10 \times 1.4286 = 14286 \]

### Step 8: Take the square root
\[ V_{\text{max}} = \sqrt{14286} \approx 119.57 \, \text{m/s} \]

### Step 9: Conclusion
The maximum safe speed of the car is approximately \( 119.57 \, \text{m/s} \).

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