Two stones A and B are projected simultaneously as shown in figure. It has been observed that both the stones reach the ground at the same place after `7` sec of their projection. Determine difference in their vertical components of initial velocities in `m//s`. `(g=9.8 m//s^(2))`

Explanation

To solve the problem, we need to determine the difference in the vertical components of the initial velocities of the two stones A and B, given that they reach the ground at the same place after 7 seconds and that the height difference between their launch points is 49 meters.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let \( u_1 \) be the initial velocity of stone A.
- Let \( u_2 \) be the initial velocity of stone B.
- Let \( \theta_1 \) be the angle of projection for stone A.
- Let \( \theta_2 \) be the angle of projection for stone B.
- The vertical component of the initial velocity for stone A is \( u_1 \sin \theta_1 \).
- The vertical component of the initial velocity for stone B is \( u_2 \sin \theta_2 \).

2. **Determine the Displacement**:
- The vertical displacement for stone A after 7 seconds is 0 (it returns to the same height).
- The vertical displacement for stone B after 7 seconds is -49 meters (it falls 49 meters).

3. **Use the Equation of Motion**:
- For stone A:
\[ y_A = u_1 \sin \theta_1 \cdot t - \frac{1}{2} g t^2 \]
Since \( y_A = 0 \) at \( t = 7 \) seconds:
\[ 0 = u_1 \sin \theta_1 \cdot 7 - \frac{1}{2} \cdot 9.8 \cdot (7)^2 \]
Rearranging gives:
\[ u_1 \sin \theta_1 = \frac{1}{2} \cdot 9.8 \cdot 7 = 34.3 \, \text{m/s} \]

- For stone B:
\[ y_B = u_2 \sin \theta_2 \cdot t - \frac{1}{2} g t^2 \]
Since \( y_B = -49 \) meters at \( t = 7 \) seconds:
\[ -49 = u_2 \sin \theta_2 \cdot 7 - \frac{1}{2} \cdot 9.8 \cdot (7)^2 \]
Rearranging gives:
\[ u_2 \sin \theta_2 = \frac{1}{2} \cdot 9.8 \cdot 7 - \frac{49}{7} \]
Simplifying:
\[ u_2 \sin \theta_2 = 34.3 + 7 = 41.3 \, \text{m/s} \]

4. **Calculate the Difference**:
- The difference in the vertical components of the initial velocities is:
\[ \Delta u = u_2 \sin \theta_2 - u_1 \sin \theta_1 \]
Substituting the values:
\[ \Delta u = 41.3 - 34.3 = 7 \, \text{m/s} \]

### Final Answer:
The difference in their vertical components of initial velocities is \( 7 \, \text{m/s} \).

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