At what angle must a railway track with a bend of 250 m radius be banked for safe running of a train at a speed of `72 kmh^(-1)`.

Explanation

To find the angle at which a railway track must be banked for safe running of a train at a given speed, we can use the formula derived from the principles of circular motion. Here’s the step-by-step solution:

### Step 1: Convert the speed from km/h to m/s
The speed of the train is given as 72 km/h. To convert this to meters per second (m/s), we use the conversion factor:
\[ \text{Speed in m/s} = \text{Speed in km/h} \times \frac{5}{18} \]
Calculating this gives:
\[ \text{Speed} = 72 \times \frac{5}{18} = 20 \text{ m/s} \]

### Step 2: Identify the radius of the bend
The radius of the bend is given as 250 m.

### Step 3: Use the formula for the banking angle
For safe turning on a banked track, the relationship between the banking angle (θ), speed (v), radius (R), and acceleration due to gravity (g) is given by:
\[ \tan \theta = \frac{v^2}{Rg} \]
Where:
- \( v = 20 \text{ m/s} \)
- \( R = 250 \text{ m} \)
- \( g = 9.8 \text{ m/s}^2 \)

### Step 4: Substitute the values into the formula
Substituting the known values into the equation:
\[ \tan \theta = \frac{(20)^2}{250 \times 9.8} \]
Calculating the right side:
\[ \tan \theta = \frac{400}{2450} \approx 0.1632 \]

### Step 5: Calculate the angle θ
To find the angle θ, we take the arctangent (inverse tangent) of 0.1632:
\[ \theta = \tan^{-1}(0.1632) \]
Calculating this gives:
\[ \theta \approx 9.27^\circ \]

### Final Answer
The angle at which the railway track must be banked for safe running of the train is approximately \( 9.27^\circ \).

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