Ball A is dropped from the top of a building of height h at the same instant that ball B is thrown vertically upward from the ground. When the balls collide, they are moving in opposite directions, and the speed of A is A times that of B. At what height does the collision occur?

To find the height at which the collision occurs, let's break down the problem step by step.

Step 1: Understand the problem.
We have two balls, A and B. Ball A is dropped vertically from the top of the building, while ball B is thrown vertically upward from the ground. When they collide, they are moving in opposite directions, and the speed of ball A is A times that of ball B. We need to find the height at which the collision occurs.

Step 2: Identify the known values.
We are given the following information:
- Ball A is dropped from the top of a building of height h.
- Ball B is thrown vertically upward from the ground.
- When the balls collide, they are moving in opposite directions.
- The speed of ball A is A times that of ball B.

Step 3: Define the variables.
Let's define the variables:
- h: height of the building
- vA: velocity of ball A
- vB: velocity of ball B
- t: time taken for the collision to occur

Step 4: Solve for the velocities.
Since ball A is dropped, its initial velocity is 0 (vA = 0). Ball B is thrown vertically upward, so we can say its initial velocity is positive (vB > 0).

Given that the speed of ball A is A times that of ball B, we can write:
vA = A * vB

Step 5: Analyze the motion of the balls.
For ball A:
Using the equations of motion, we can find the distance covered by ball A when it collides with ball B. The equation is:
h = 0.5 * g * t^2, where g is the acceleration due to gravity.

For ball B:
Since ball B is thrown upward, it will reach its maximum height when its velocity becomes zero. We can use this fact to find the time taken for ball B to reach its maximum height.
0 = vB - g * t, where g is the acceleration due to gravity.

Step 6: Solve the equations.
Let's solve the equations to find the values of vB and t.

From the equation vA = A * vB, we have:
0 = A * vB - g * t

From 0 = vB - g * t, we have:
vB = g * t

Substituting the value of vB in the first equation, we get:
0 = A * (g * t) - g * t
0 = g * t * (A - 1)

Since t cannot be zero, we can cancel g from both sides of the equation:
0 = A - 1

Therefore, A = 1.

This implies that the speed of ball A is equal to the speed of ball B. As a result, they will never collide.

Therefore, there is no specific height at which the collision occurs since the collision does not occur.

Note: It's essential to carefully analyze the problem to ensure that the given information is consistent and can lead to a solution. In this case, we determined that the collision does not occur based on the given conditions.