Two balls `A and B` of same masses are thrown from the top of the building `A`. Thrown upward with velocity `V and B`, thrown downward with velocity `V`, then

Options

A. (a)Velocity of A is more than B at the ground
B. (b)Velocity of B is more than A at the ground
C. (c)Both A and B strike the ground with same velocity
D. (d)None of these

Explanation

To solve the problem of two balls A and B being thrown from the top of a building, we will analyze their motions step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Ball A is thrown upward with an initial velocity \( V \).
- Ball B is thrown downward with the same initial velocity \( V \).
- Both balls are thrown from the same height (the top of the building).

2. **Analyzing Ball A (Thrown Upward)**:
- When ball A is thrown upward, it will first rise to a maximum height before falling back down.
- The time taken to reach the maximum height can be calculated using the formula:
\[ t_{up} = \frac{V}{g} \]
where \( g \) is the acceleration due to gravity.

3. **Calculating Maximum Height for Ball A**:
- The maximum height \( h_{max} \) reached by ball A can be calculated using the equation:
\[ h_{max} = \frac{V^2}{2g} \]

4. **Total Height from which Ball A Falls**:
- The total height from which ball A falls to the ground is the height of the building \( h \) plus the maximum height it reached:
\[ H_A = h + h_{max} = h + \frac{V^2}{2g} \]

5. **Analyzing Ball B (Thrown Downward)**:
- Ball B is thrown downward with the same initial velocity \( V \).
- The time taken for ball B to hit the ground can be calculated using the equation of motion:
\[ H_B = h + \frac{V^2}{2g} \]

6. **Using the Equation of Motion**:
- For both balls, we can use the equation of motion to find the final velocity just before they hit the ground:
\[ v^2 = u^2 + 2gH \]
- For ball A:
\[ v_A^2 = V^2 + 2g\left(h + \frac{V^2}{2g}\right) = V^2 + 2gh + V^2 = 2V^2 + 2gh \]
- For ball B:
\[ v_B^2 = V^2 + 2g\left(h\right) = V^2 + 2gh \]

7. **Final Velocities**:
- Both balls will have the same final velocity when they hit the ground:
\[ v_A = v_B \]

8. **Conclusion**:
- Since both balls hit the ground with the same velocity and are thrown from the same height, they will also hit the ground at the same time.