From a building two balls `A` and `B` are thrown such that `A` is thrown upwards and `B` downwards ( both vertically with the same speed ). If `v_(A) and v_(B)` are their respective velocities on reaching the ground , then

Options

A. ` v_(B) gt v_(A)`
B. ` v_(A) = v_(B)`
C. ` v_(A) gt v_(B)`
D. their velocities depend on their masses.

Explanation

To solve the problem of two balls, A and B, being thrown from a building with the same initial speed but in opposite directions, we need to analyze their motions and determine their velocities when they reach the ground.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- Ball A is thrown upwards with an initial velocity \( u \).
- Ball B is thrown downwards with the same initial velocity \( u \).
- Both balls are thrown from the same height \( H \).

**Hint**: Identify the initial velocities and directions of both balls.

2. **Analyzing Ball A's Motion**:
- When Ball A is thrown upwards, it will first ascend until it reaches its highest point, where its velocity will be \( 0 \).
- After reaching the highest point, it will start descending back towards the ground.
- The total distance it travels upwards is \( H + h \), where \( h \) is the height it rises above the initial point before descending.
- The final velocity \( v_A \) when it reaches the ground can be calculated using the equation of motion:
\[ v_A^2 = u^2 + 2gh \]
- Since it will fall back down through the height \( H \), we can consider the total height \( H + h \) for the downward motion.

3. **Analyzing Ball B's Motion**:
- Ball B is thrown directly downwards with the same initial velocity \( u \).
- The distance it travels to reach the ground is \( H \).
- The final velocity \( v_B \) when it reaches the ground can be calculated using the same equation of motion:
\[ v_B^2 = u^2 + 2gH \]

4. **Comparing the Final Velocities**:
- For Ball A, after reaching the highest point and then falling back down, the total distance it falls is \( H + h \).
- For Ball B, the total distance it falls is \( H \).
- However, due to the symmetry of the motion, both balls will have the same final speed when they reach the ground.
- Thus, we can conclude that:
\[ v_A = v_B \]

5. **Conclusion**:
- The final velocities of both balls when they reach the ground are equal, i.e., \( v_A = v_B \).

### Final Answer:
Both balls will have the same velocity upon reaching the ground: \( v_A = v_B \).