How does the horizontal distance a projectile falls below an otherwise straight line path compare with the vertical distance it would fall from rest in the same time?

The horizontal distance a projectile falls below an otherwise straight-line path is equal to the vertical distance it would fall from rest in the same time due to the independence of horizontal and vertical motions. In projectile motion, the horizontal component moves at a constant speed while the vertical component is influenced by gravity. Therefore, the vertical drop in a given time interval is determined solely by the acceleration due to gravity, which is the same regardless of the horizontal motion. As a result, the horizontal and vertical displacements can be analyzed separately, leading to this direct comparison.