A person standing near the edge of the top of a building throws two balls A and B. The ball A is thrown vertically upward and B is thrown vertically downward with the same speed. The ball A hits the ground with speed `v_A` and the ball B hits the ground wiht a speed `v_B`. We have

Options

A. `v_Altv_B`
B. `v_Altv_B`
C. `v_A=v_B`
D. the relation between `v_A and v_B` depends on height of the building above the ground.

Answer

To solve the problem, we need to analyze the motion of both balls A and B as they are thrown from the top of a building.

### Step-by-Step Solution:

1. **Understanding the Situation:**
- A person throws two balls from the top of a building: Ball A is thrown vertically upward and Ball B is thrown vertically downward with the same initial speed (let's denote this speed as \( u \)).
- Both balls are thrown from the same height \( H \) (the height of the building).

2. **Analyzing Ball A (Thrown Upward):**
- When Ball A is thrown upward, it will first move against gravity, reach a maximum height, and then fall back down to the ground.
- The final speed of Ball A when it hits the ground can be calculated using the kinematic equation:
\[ v_A^2 = u^2 + 2gh \]
Here, \( g \) is the acceleration due to gravity (acting downward), and \( h \) is the total height from which it falls (which is \( H \)).

3. **Analyzing Ball B (Thrown Downward):**
- Ball B is thrown downward with the same initial speed \( u \).
- The final speed of Ball B when it hits the ground can also be calculated using the same kinematic equation:
\[ v_B^2 = u^2 + 2gh \]
The displacement is the same \( H \) since it is also falling from the same height.

4. **Comparing the Final Speeds:**
- Since both balls are thrown with the same initial speed \( u \) and fall the same distance \( H \), we can equate the two equations:
\[ v_A^2 = u^2 + 2gH \]
\[ v_B^2 = u^2 + 2gH \]
- From these equations, we can see that:
\[ v_A^2 = v_B^2 \]
- Taking the square root of both sides gives us:
\[ v_A = v_B \]

5. **Conclusion:**
- Therefore, the final speeds of both balls when they hit the ground are equal:
\[ v_A = v_B \]

### Final Answer:
The relation between \( v_A \) and \( v_B \) is:
\[ v_A = v_B \]