An aeroplane is flying in a horizontal direction with a velocity of 900 km/h and at a height of 1960m. When it is vertically above the point A on the ground, a body is dropped from it. The body strikes the ground at point B. The distance AB will be (take `g = 9.8 m//s ^(2))`

Options

A. `3.33 km`
B. `5km`
C. `10km`
D. `18km`

Answer

To solve the problem, we need to find the horizontal distance \( AB \) that the body travels while it falls from the airplane to the ground. Here’s how we can do it step by step:

### Step 1: Convert the velocity of the airplane from km/h to m/s
The velocity of the airplane is given as \( 900 \, \text{km/h} \). We need to convert this to meters per second (m/s) using the conversion factor \( 1 \, \text{km/h} = \frac{1}{3.6} \, \text{m/s} \).

\[ \text{Velocity in m/s} = 900 \times \frac{1}{3.6} = 250 \, \text{m/s} \]

### Step 2: Calculate the time taken for the body to fall to the ground
The height from which the body is dropped is \( 1960 \, \text{m} \). We can use the second equation of motion to find the time \( t \) it takes for the body to fall. The equation is:

\[ S = ut + \frac{1}{2} g t^2 \]

Where:
- \( S = 1960 \, \text{m} \) (the height)
- \( u = 0 \, \text{m/s} \) (initial velocity in the vertical direction)
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)

Substituting the values into the equation, we have:

\[ 1960 = 0 \cdot t + \frac{1}{2} \cdot 9.8 \cdot t^2 \]

This simplifies to:

\[ 1960 = 4.9 t^2 \]

### Step 3: Solve for \( t^2 \)
Rearranging the equation gives:

\[ t^2 = \frac{1960}{4.9} \approx 400 \]

Taking the square root:

\[ t = \sqrt{400} = 20 \, \text{s} \]

### Step 4: Calculate the horizontal distance \( AB \)
Now that we have the time \( t \), we can find the horizontal distance \( AB \) using the formula:

\[ \text{Distance} = \text{Velocity} \times \text{Time} \]

Substituting the values we have:

\[ AB = 250 \, \text{m/s} \times 20 \, \text{s} = 5000 \, \text{m} \]

### Step 5: Convert the distance to kilometers
To convert meters to kilometers:

\[ AB = \frac{5000}{1000} = 5 \, \text{km} \]

### Final Answer
The distance \( AB \) is \( 5 \, \text{km} \).

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