A stone is dropped from a height h. simultaneously, another stone is thrown up from the ground which reaches a height 4h. The two stones will cross each other after time:-

Options

A. `sqrt((h)/(8g))`
B. `sqrt(8gh)`
C. `sqrt(2gh)`
D. `sqrt((h)/(2g))`

Explanation

To solve the problem step by step, we will analyze the motion of both stones and find the time at which they cross each other.

### Step 1: Understand the motion of both stones
- **Stone A** is dropped from a height \( h \).
- **Stone B** is thrown upwards from the ground and reaches a maximum height of \( 4h \).

### Step 2: Determine the initial velocity of Stone B
At the maximum height of Stone B (which is \( 4h \)), its final velocity is \( 0 \). We can use the kinematic equation:
\[ v^2 = u^2 - 2gh \]
where:
- \( v = 0 \) (final velocity at the maximum height)
- \( u \) is the initial velocity of Stone B
- \( h = 4h \) (the height it reaches)

Substituting the values:
\[ 0 = u^2 - 2g(4h) \]
This simplifies to:
\[ u^2 = 8gh \]
Thus, the initial velocity \( u \) of Stone B is:
\[ u = \sqrt{8gh} \]

### Step 3: Analyze the motion of both stones
- **Stone A** (dropped from height \( h \)):
- Initial velocity \( u_A = 0 \)
- Distance fallen after time \( t \):
\[ s_A = \frac{1}{2}gt^2 \]

- **Stone B** (thrown upwards):
- Initial velocity \( u_B = \sqrt{8gh} \)
- Distance traveled upwards after time \( t \):
\[ s_B = u_B t - \frac{1}{2}gt^2 = \sqrt{8gh} t - \frac{1}{2}gt^2 \]

### Step 4: Set up the equation for when the stones cross each other
The stones will cross each other when the sum of the distances they have traveled equals the total height (which is \( h + 4h = 5h \)):
\[ s_A + s_B = 5h \]
Substituting the expressions for \( s_A \) and \( s_B \):
\[ \frac{1}{2}gt^2 + \left(\sqrt{8gh} t - \frac{1}{2}gt^2\right) = 5h \]
This simplifies to:
\[ \sqrt{8gh} t = 5h \]

### Step 5: Solve for time \( t \)
Rearranging gives:
\[ t = \frac{5h}{\sqrt{8gh}} = \frac{5\sqrt{h}}{\sqrt{8g}} = \frac{5}{\sqrt{8g}} \sqrt{h} \]

### Final Answer
The time at which the two stones cross each other is:
\[ t = \frac{5\sqrt{h}}{\sqrt{8g}} \]