From a high tower, at time `t=0`, one stone is dropped from rest and simultaneously another stone is projected vertically up with an initial velocity .The graph of distance `S` between the two stones plotted against time `t` will be

Options

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B. .
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D. .

Explanation

To solve the problem, we need to analyze the motion of the two stones and determine the distance \( S \) between them as a function of time \( t \).

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- Stone 1 is dropped from rest from a height \( h \) at time \( t = 0 \).
- Stone 2 is projected upwards with an initial velocity \( u \) at the same time.

2. **Determine the Distance Traveled by Each Stone**:
- For Stone 1 (dropped from rest):
- The distance \( S_1 \) it falls after time \( t \) can be calculated using the equation of motion:
\[ S_1 = \frac{1}{2} g t^2 \]
where \( g \) is the acceleration due to gravity.

- For Stone 2 (projected upwards):
- The distance \( S_2 \) it travels upwards after time \( t \) can be calculated as:
\[ S_2 = ut - \frac{1}{2} g t^2 \]
Here, \( ut \) is the distance it would travel upwards without gravity, and \( \frac{1}{2} g t^2 \) accounts for the downward acceleration due to gravity.

3. **Calculate the Distance Between the Two Stones**:
- The distance \( S \) between the two stones at any time \( t \) is given by:
\[ S = h - S_1 - S_2 \]
- Substituting the expressions for \( S_1 \) and \( S_2 \):
\[ S = h - \left(\frac{1}{2} g t^2\right) - \left(ut - \frac{1}{2} g t^2\right) \]
- Simplifying this expression:
\[ S = h - ut + g t^2 \]

4. **Determine the Nature of the Graph**:
- The equation \( S = h - ut + g t^2 \) is a quadratic equation in \( t \) since it can be rearranged to:
\[ S = -\frac{1}{2} g t^2 + ut + h \]
- This indicates that the graph of distance \( S \) against time \( t \) will be a parabola opening downwards (since the coefficient of \( t^2 \) is negative).

5. **Conclusion**:
- The graph of the distance \( S \) between the two stones plotted against time \( t \) will be a downward-opening parabola.