An aeroplane executes a horizontal loop at a speed of 720 kmph with its wings banked at 45°. What is the radius of the loop? (Take g = 10 `ms^(-2)` )

Options

A. 7.2 km
B. 4 km
C. 2 km
D. 4.5 km

Explanation

To solve the problem of finding the radius of the loop executed by the aeroplane, we will follow these steps:

### Step 1: Convert the speed from km/h to m/s
The speed of the aeroplane is given as 720 km/h. To convert this to meters per second (m/s), we use the conversion factor \( \frac{5}{18} \).

\[ \text{Speed in m/s} = 720 \times \frac{5}{18} = 200 \, \text{m/s} \]

### Step 2: Use the formula for the banking angle
The banking angle \( \theta \) is related to the speed \( v \), radius \( r \), and acceleration due to gravity \( g \) by the formula:

\[ \tan(\theta) = \frac{v^2}{rg} \]

Given that \( \theta = 45^\circ \), we know that \( \tan(45^\circ) = 1 \).

### Step 3: Set up the equation
Substituting the known values into the equation:

\[ 1 = \frac{(200)^2}{r \cdot 10} \]

### Step 4: Solve for the radius \( r \)
Rearranging the equation to solve for \( r \):

\[ r = \frac{(200)^2}{10} \]

Calculating the right side:

\[ r = \frac{40000}{10} = 4000 \, \text{m} \]

### Step 5: Convert the radius to kilometers
Since the radius is often expressed in kilometers, we convert meters to kilometers:

\[ r = 4000 \, \text{m} = 4 \, \text{km} \]

### Final Answer
The radius of the loop is **4 km**.

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