1. Given: Two balls are thrown vertically from the top of the same building:

   • Ball $A$ is thrown upward with speed $u$
   • Ball $B$ is thrown downward with speed $u$

   Both reach the ground with velocities $v_A$ and $v_B$ respectively.

2. Use kinematic relation

   For vertical motion under gravity, $v^2 = u^2 + 2as$

   Take downward direction as positive.

   Let the height of the building be $h$.

3. For ball $A$:

   • Initial velocity is upward, so $u_A = -u$
   • Acceleration due to gravity: $a = g$
   • Displacement to ground: $s = h$

   Therefore, $v_A^2 = (-u)^2 + 2gh = u^2 + 2gh$

4. For ball $B$:

   • Initial velocity is downward, so $u_B = +u$
   • Acceleration: $a = g$
   • Displacement: $s = h$

   Therefore, $v_B^2 = u^2 + 2gh$

5. Compare the two: $v_A^2 = v_B^2$ Since both balls reach the ground moving downward, their velocities are equal in magnitude and direction.

   Hence, $v_A = v_B$

6. Check options:

   • A: ${v_B} > {v_A}$ → False
   • B: ${v_A} = {v_B}$ → True
   • C: ${v_A} > {v_B}$ → False
   • D: Depends on masses → False, because neglecting air resistance, mass does not affect the result.

Final Answer: Option B.