A circular race track of radius 120 m is banked at an angle of `53^o` . The optimum speed of car to avoid wear and tear of its tyres is (g = 10 ms^-2)

Options

A. 40 m/s
B. 20 m/s
C. 30 m/s
D. 10 m/s

Answer

To find the optimum speed of a car on a banked circular track, we can use the formula for the optimum speed \( v \):

\[ v = \sqrt{rg \tan \theta} \]

where:
- \( r \) is the radius of the track,
- \( g \) is the acceleration due to gravity,
- \( \theta \) is the banking angle.

Given:
- \( r = 120 \, \text{m} \)
- \( g = 10 \, \text{m/s}^2 \)
- \( \theta = 53^\circ \)

### Step 1: Calculate \( \tan \theta \)

First, we need to find \( \tan 53^\circ \).

Using trigonometric values, we know:
\[ \tan 53^\circ \approx 1.6 \quad (\text{or more precisely, } \tan 53^\circ = \frac{4}{3}) \]

### Step 2: Substitute the values into the formula

Now, we can substitute the values into the formula:

\[ v = \sqrt{120 \times 10 \times \tan 53^\circ} \]

Substituting \( \tan 53^\circ = \frac{4}{3} \):

\[ v = \sqrt{120 \times 10 \times \frac{4}{3}} \]

### Step 3: Simplify the expression

Now, simplify the expression inside the square root:

\[ v = \sqrt{1200 \times \frac{4}{3}} = \sqrt{1200 \times 1.3333} = \sqrt{1600} \]

### Step 4: Calculate the square root

Calculating the square root gives us:

\[ v = 40 \, \text{m/s} \]

### Conclusion

Thus, the optimum speed of the car to avoid wear and tear of its tires is:

\[ \boxed{40 \, \text{m/s}} \]

---