From certain height 'h' two bodies are projected horizontally each with velocity v. One body is projected towards North and the other body is projected towards east. Their separation on reaching the ground.

Options

A. `sqrt((2v^2h)/(g))`
B. `sqrt((4v^2 h)/(g))`
C. `sqrt((v^2 h)/(g))`
D. `sqrt((8 v^2 h)/(g))`

Answer

To find the separation between the two bodies projected horizontally from a height \( h \), we can follow these steps:

### Step 1: Determine the time of flight
When a body is projected horizontally from a height \( h \), the time \( t \) it takes to reach the ground can be calculated using the formula for free fall:
\[ t = \sqrt{\frac{2h}{g}} \]
where \( g \) is the acceleration due to gravity.

### Step 2: Calculate the horizontal distance traveled by each body
Both bodies are projected horizontally with the same velocity \( v \). The horizontal distance \( r \) traveled by each body when they reach the ground is given by:
\[ r = v \cdot t \]
Substituting the expression for \( t \):
\[ r = v \cdot \sqrt{\frac{2h}{g}} \]

### Step 3: Determine the separation between the two bodies
Since one body is projected towards the North and the other towards the East, they will form a right triangle with the horizontal distances as the two legs. The separation \( s \) between the two bodies when they reach the ground can be calculated using the Pythagorean theorem:
\[ s = \sqrt{(r)^2 + (r)^2} = \sqrt{2r^2} = r\sqrt{2} \]
Substituting the expression for \( r \):
\[ s = \sqrt{2} \cdot \left(v \cdot \sqrt{\frac{2h}{g}}\right) = v \cdot \sqrt{\frac{2h}{g}} \cdot \sqrt{2} = v \cdot \frac{2\sqrt{h}}{\sqrt{g}} \]

### Step 4: Final expression for separation
Thus, the final expression for the separation \( s \) becomes:
\[ s = \frac{2v\sqrt{h}}{\sqrt{g}} \]

### Step 5: Simplifying the expression
Alternatively, we can express it as:
\[ s = \frac{4hv^2}{g} \]

### Final Result
The separation between the two bodies when they reach the ground is:
\[ s = \frac{4hv^2}{g} \]