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Projectile motion is a fascinating topic in physics that deals with the motion of objects thrown or projected into the air, influenced solely by gravity. When a projectile is launched, it follows a parabolic path known as its trajectory. This path can be broken down into horizontal and vertical components. At the highest point of its trajectory, the vertical component of its velocity is zero, leaving only the horizontal component involved, which remains constant throughout the flight (assuming no air resistance). This scenario is important in understanding what happens at the peak of a projectile's flight, which in many problems, like the one described, is when interesting events such as explosions may occur.

Horizontal velocity refers to the constant speed at which a projectile travels horizontally. Once an object is in motion, and ignoring air resistance, this velocity doesn't change. It becomes crucial when the object reaches its highest point, as its vertical velocity momentarily drops to zero. In our exercise, the projectile, when it reaches its highest point, moves with horizontal velocity only. Upon the explosion, this horizontal velocity is distributed among the resulting fragments, affecting their subsequent trajectory. In this case, each fragment gains an equal but opposite horizontal velocity, relative to the projectile, thereby conserving the overall momentum.

Explosion physics in the context of projectiles is an application of momentum conservation. An explosion is simply a rapid increase in volume with intense energy release, causing fragments to move apart. Each fragment moves such that the total momentum of the system remains unchanged. In our scenario, the projectile—being at the peak of its trajectory—explodes into two fragments, each with equal and opposite horizontal velocities. This means the overall horizontal momentum before and after the explosion remains the same. The fragment landing back at point A exhibits how the explosion effectively reversed the velocity vector, creating balance for the motion of the exploaded parts.

Calculating the horizontal distance a projectile—or its fragments—covers involves understanding its velocity and the time of travel. For a projectile that has exploded into fragments, the conservation of momentum guides us to realize how the distances relate. One fragment ends up returning to point A after covering a distance \( -D \); thus, the second fragment must cover the remainder of the horizontal momentum assigned distance. From a restitution framework, the second fragment moves an additional amount equivalent to gaining twice the original horizontal velocity, landing it at a total of \( 3D \) from the start point. This calculation underscores how interplay of velocities and time dictate the landing spots post-explosion.