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 using the equations of motion under gravity. Let's denote:

- \( U \): the initial speed at which both balls are thrown (same for both).
- \( g \): the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).
- \( h \): the height of the building.
- \( v_A \): the final speed of ball A when it hits the ground.
- \( v_B \): the final speed of ball B when it hits the ground.

### Step 1: Analyze the motion of ball A
Ball A is thrown vertically upward. When it reaches its maximum height, its velocity becomes zero. After that, it falls back down to the ground. We can use the following equation of motion to find the speed of ball A when it hits the ground:

\[ v_A^2 = U^2 - 2gh \]

Here, the displacement is negative because the ball is moving upward first and then downward, but we will consider the total height it falls from the maximum height back to the ground.

### Step 2: Analyze the motion of ball B
Ball B is thrown vertically downward. We can use the same equation of motion to find the speed of ball B when it hits the ground:

\[ v_B^2 = U^2 + 2gh \]

Here, the displacement is positive since the ball is thrown downward.

### Step 3: Compare the speeds \( v_A \) and \( v_B \)
Now we can compare the two equations:

1. From the equation for ball A:
\[ v_A^2 = U^2 - 2gh \]

2. From the equation for ball B:
\[ v_B^2 = U^2 + 2gh \]

### Step 4: Determine the relationship between \( v_A \) and \( v_B \)
To compare \( v_A \) and \( v_B \), we can see that:

- \( v_A^2 < v_B^2 \) because \( -2gh < +2gh \).
- Taking the square root of both sides (keeping in mind that both speeds are positive), we conclude that:

\[ v_A < v_B \]

### Conclusion
Thus, the relationship between the speeds of the two balls when they hit the ground is:

\[ v_A < v_B \]