A park has a radis of 10 m. If a vehicle goes round it at an averge speed of 18 km/hr, what shoud be the proper angle of banking?

Explanation

To solve the problem of finding the proper angle of banking for a vehicle going around a circular path, we can follow these steps:

### Step 1: Convert the speed from km/hr to m/s
The speed of the vehicle is given as 18 km/hr. To convert this to meters per second (m/s), we use the conversion factor:
\[ \text{Speed in m/s} = \text{Speed in km/hr} \times \frac{5}{18} \]
So,
\[ \text{Speed} = 18 \times \frac{5}{18} = 5 \text{ m/s} \]

### Step 2: Identify the radius of the circular path
The radius \( R \) of the park is given as 10 m.

### Step 3: Use the formula for the angle of banking
The angle of banking \( \theta \) can be calculated using the formula:
\[ \tan \theta = \frac{V^2}{Rg} \]
where:
- \( V \) is the speed of the vehicle (5 m/s),
- \( R \) is the radius of the circular path (10 m),
- \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)).

### Step 4: Substitute the values into the formula
Now we substitute the known values into the formula:
\[ \tan \theta = \frac{(5)^2}{10 \times 10} \]
Calculating this gives:
\[ \tan \theta = \frac{25}{100} = \frac{1}{4} \]

### Step 5: Calculate the angle \( \theta \)
To find the angle \( \theta \), we take the inverse tangent (arctan) of \( \frac{1}{4} \):
\[ \theta = \tan^{-1}\left(\frac{1}{4}\right) \]

### Step 6: Final Calculation
Using a calculator, we find:
\[ \theta \approx 14.04^\circ \]

Thus, the proper angle of banking for the vehicle is approximately \( 14.04^\circ \).

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