Two stones of equal masses are dropped from a rooftop of height h one after another. Their separation distance against time will

Options

A. Remain the same
B. Increase
C. Decrease
D. Be zero

Explanation

To find the separation distance between two stones dropped from a rooftop of height \( h \) one after another, we can follow these steps:

### Step 1: Understand the Problem
Two stones are dropped from the same height \( h \) but at different times. We need to analyze how the separation distance between the two stones changes over time.

### Step 2: Define Variables
- Let the first stone be dropped at time \( t = 0 \).
- Let the second stone be dropped at time \( t = t_0 \) (after a delay).
- Let \( g \) be the acceleration due to gravity.

### Step 3: Calculate the Distance Fallen by Each Stone
Using the second equation of motion, the distance fallen by an object under gravity is given by:
\[ s = ut + \frac{1}{2} a t^2 \]
Since both stones are dropped (initial velocity \( u = 0 \)), the distance fallen by the first stone after time \( t \) is:
\[ H_1 = \frac{1}{2} g t^2 \]
For the second stone, which is dropped at \( t_0 \) seconds later, the time it has been falling when the first stone has been falling for \( t \) seconds is \( t - t_0 \). Therefore, the distance fallen by the second stone is:
\[ H_2 = \frac{1}{2} g (t - t_0)^2 \]

### Step 4: Calculate the Separation Distance
The separation distance \( \Delta H \) between the two stones at any time \( t \) is given by:
\[ \Delta H = H_1 - H_2 \]
Substituting the expressions for \( H_1 \) and \( H_2 \):
\[ \Delta H = \frac{1}{2} g t^2 - \frac{1}{2} g (t - t_0)^2 \]

### Step 5: Expand the Expression
Expanding \( (t - t_0)^2 \):
\[ (t - t_0)^2 = t^2 - 2tt_0 + t_0^2 \]
Thus,
\[ H_2 = \frac{1}{2} g (t^2 - 2tt_0 + t_0^2) = \frac{1}{2} g t^2 - g tt_0 + \frac{1}{2} g t_0^2 \]
Now substituting back into the separation distance:
\[ \Delta H = \frac{1}{2} g t^2 - \left(\frac{1}{2} g t^2 - g tt_0 + \frac{1}{2} g t_0^2\right) \]
This simplifies to:
\[ \Delta H = g tt_0 - \frac{1}{2} g t_0^2 \]

### Step 6: Analyze the Result
The expression for separation distance \( \Delta H \) is:
\[ \Delta H = g t t_0 - \frac{1}{2} g t_0^2 \]
This shows that as time \( t \) increases, the term \( g t t_0 \) increases linearly with \( t \), while the term \( -\frac{1}{2} g t_0^2 \) is constant. Therefore, the overall separation distance \( \Delta H \) increases with time.

### Conclusion
The separation distance between the two stones increases with time.

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