Your approach is way off, and you really don't need to use vectors for this.

First find the distance traveled by the first ball:

$$d_A(t) = a_At^2+v_At$$

then find the distance traveled by the second ball:

$$d_B(t) = a_Bt^2+v_Bt$$

You know that the two balls are pointed in opposite directions, so think about the signs of the accelerations $a$ and the velocities $v$ (i.e. does acceleration add to the velocity or work against it?).

Then once you get each distance, you can simply add them together.

What you were doing is trying to find some kind of "average or resultant velocity" for the two objects, which is not physically meaningful.

Your approach is way off, and you really don't need to use vectors for this.

First find the distance traveled by the first ball:

$$d_A(t) = a_At^2+v_At$$

then find the distance traveled by the second ball:

$$d_B(t) = a_Bt^2+v_Bt$$

You know that the two balls are pointed in opposite directions, so think about the signs of the accelerations $a$ and the velocities $v$ (i.e. does acceleration add to the velocity or work against it?).

Then once you get each distance, you can simply add them together.

What you were doing is trying to find some kind of "average or resultant velocity" for the two objects, which is not physically meaningful.

Also, in general rather than trying to find the correct formula for a physics problem, it's best to examine what is physically happening, especially by drawing diagrams, and determine the relationships between quantities (in this case, how does distance traveled related to starting velocity, acceleration, and time elapsed), until you find a way to compute what is needed from what is known. No formula will work if it is applied incorrectly.