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Understanding the work done by gravity during projectile motion is critical in physics. It's a force we constantly experience and it's always acting downwards toward the center of the Earth. When a projectile is fired horizontally, like in our exercise, gravity starts to pull it down immediately.

The work done by gravity on an object, such as our projectile, is the product of three things: the force of gravity on the object (\(mg\)), the displacement in the direction of the force (\(h\)), and the cosine of the angle between them. Since gravity acts vertically and the displacement we're considering is also vertical, the angle between the force and the displacement is 0 degrees, and the cosine of 0 degrees is 1.

So, the work done by gravity on the projectile, denoted as \(W_{g}\), is simply \(W_{g} = m \times g \times h\). This calculation assumes no air resistance and a flat terrain. Without these simplifications, the scenario would be significantly more complex.

Taking a closer look at kinetic energy change helps us understand an object's motion dynamics. Kinetic energy (\(KE\)) is the energy of motion and is given by the formula \(KE = \frac{1}{2}mv^2\). The change in kinetic energy, denoted as \(\Delta KE\), is the final kinetic energy minus the initial kinetic energy.

In the context of our projectile, it starts with a certain amount of kinetic energy due to its initial speed \(v_0\), calculated as \(\frac{1}{2} m v_0^2\). As it falls under the influence of gravity, its vertical speed increases, changing its overall kinetic energy.

The fascinating part is that, despite the absence of air resistance and assuming a purely horizontal launch, the change in kinetic energy exactly equals the work done by gravity. This beautifully demonstrates the conservation of energy – a key principle in physics, showing that all the gravitational work done on the projectile ends up as an addition to its kinetic energy.

The work-energy theorem is a fundamental principle that connects force, work, and an object's kinetic energy. It states that the total work done by all forces acting on an object is equal to the change in its kinetic energy. This theorem is beautifully illustrated in our example of projectile motion.

When applying the work-energy theorem to our projectile, we see that the only force doing work is gravity (\(W_{g}\)). Therefore, the change in the projectile's kinetic energy must be identical to the work done by gravity. By comparing the kinetic energy at launch and just before impact, we're able to see the direct impact of gravity's work on the object's kinetic energy.

The theorem also reaffirms that energy in a system is conserved. In the absence of dissipative forces like air resistance, all the energy from gravity's pull translates into kinetic energy. This principle also highlights why our answers remain the same regardless of the projectile's launch angle – because the work done by gravity is dependent only on the vertical height through which the projectile falls, not on the path it takes.