For traffic moving at 60 km/hour along a circular track of radius 0.1 km, the correct angle of banking is

Options

A. `{(60)^(2)// 0.1 }` km
B. `tan^(-1) [((50//3)^(2))/(100 xx 9.8)]`
C. `tan^(-1) [(100 xx 9.8)/((50//3)^(2))]`
D. `tan^(-1) sqrt((60 xx 0.1 xx 9.8))`

Explanation

To find the correct angle of banking for traffic moving at 60 km/hour along a circular track of radius 0.1 km, we can use the formula for the banking angle:

\[ \theta = \tan^{-1}\left(\frac{V^2}{Rg}\right) \]

where:
- \(V\) = velocity of the vehicle in meters per second (m/s)
- \(R\) = radius of the circular track in meters (m)
- \(g\) = acceleration due to gravity (approximately \(9.8 \, \text{m/s}^2\))

### Step 1: Convert the speed from km/hour to m/s
Given speed = 60 km/hour.

To convert km/hour to m/s, we use the conversion factor:

\[ 1 \, \text{km/hour} = \frac{1000 \, \text{meters}}{3600 \, \text{seconds}} = \frac{1}{3.6} \, \text{m/s} \]

So,

\[ V = 60 \times \frac{1}{3.6} = \frac{60}{3.6} = \frac{600}{36} = \frac{50}{3} \, \text{m/s} \]

### Step 2: Convert the radius from km to m
Given radius = 0.1 km.

To convert km to m:

\[ R = 0.1 \times 1000 = 100 \, \text{m} \]

### Step 3: Substitute values into the formula
Now we can substitute \(V\), \(R\), and \(g\) into the formula for the banking angle:

\[ \theta = \tan^{-1}\left(\frac{V^2}{Rg}\right) \]

Substituting the values:

\[ \theta = \tan^{-1}\left(\frac{\left(\frac{50}{3}\right)^2}{100 \times 9.8}\right) \]

Calculating \(V^2\):

\[ \left(\frac{50}{3}\right)^2 = \frac{2500}{9} \]

Now substituting this into the equation:

\[ \theta = \tan^{-1}\left(\frac{\frac{2500}{9}}{100 \times 9.8}\right) \]

Calculating the denominator:

\[ 100 \times 9.8 = 980 \]

So the equation becomes:

\[ \theta = \tan^{-1}\left(\frac{2500}{9 \times 980}\right) \]

Calculating \(9 \times 980\):

\[ 9 \times 980 = 8820 \]

Thus, we have:

\[ \theta = \tan^{-1}\left(\frac{2500}{8820}\right) \]

### Step 4: Calculate the angle
Now we can compute the value:

\[ \frac{2500}{8820} \approx 0.283 \]

Using a calculator to find the arctan:

\[ \theta \approx \tan^{-1}(0.283) \approx 15.8^\circ \]

### Final Answer
The correct angle of banking is approximately \(15.8^\circ\).

---