If one ball is dropped and another is thrown downward from the same height, their arrival times at the ground depend on the initial velocity of the thrown ball. If the thrown ball has a downward velocity, it will reach the ground faster than the dropped ball. However, if both are released from rest, they will hit the ground simultaneously due to gravity acting equally on both. The discussion emphasizes understanding basic mechanics principles to analyze the scenario accurately. Overall, the outcome hinges on the initial conditions of the balls' motion.