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Projectile motion describes the movement of objects thrown or fired into the air, like a ball or, in this exercise, a chicken-sized object. These objects follow a curved path due to gravitational forces. The key to understanding projectile motion is to break it into two components: horizontal and vertical motion.

The horizontal motion remains constant because, under ideal conditions, there is no acceleration in this direction. In the vertical direction, the only force acting is gravity, leading to a constant acceleration of approximately 9.81 m/s² downward.

This problem involves finding the projectile's speed and distance at a specific time, which combines these concepts. Here, the gun accelerates the projectile until it leaves the barrel, forming part of the preliminary phase before true projectile motion takes place.

In this exercise, Newton's Laws of Motion play a central role in determining the forces acting on the projectile. Newton's Second Law of Motion states that force equals mass times acceleration, or \( F = ma \).

This law is crucial when calculating the net forces acting on the object at different instance times. Initially, a higher force is needed to overcome inertia and accelerate the object until it exits the barrel, reaching a high speed. As calculated in the solution, the force required varies with time depending on the acceleration observed at that specific moment.

Newton's second law connects directly to this change in forces, highlighting that a larger force is necessary at the initial stages of motion to achieve the desired velocity in a short time.

Calculus is a powerful tool in physics for deriving equations of motion and resolving rates of change. This exercise involves using calculus to find velocity and acceleration from displacement.

The displacement function, \( x(t) \), given as a function of time, allows us to derive velocity by differentiating once with respect to time. The result is the velocity function which reveals the speed at any moment.

Further, differentiating the velocity function gives the acceleration, a key factor reflecting how fast the velocity of the object increases with time. With these derivative functions, not only can we find instantaneous velocities and accelerations, but they enable us to apply Newton's laws effectively to determine forces.

Kinematics deals with the description of motion without considering the forces causing it. It involves variables like displacement, velocity, and acceleration.

In kinematics, understanding equations that define motion over time is key. Here, the given displacement equation serves as a starting point to unravel a story of movement. By employing calculus, students can transition from this equation to derive velocity and acceleration functions, thus gaining insight into how motion progresses.

This method is fundamental in physics to predict motion paths and outcomes without external force considerations initially. By embracing the principles of kinematics, one can closely examine each stage of an object's journey, from rest to motion, in a very descriptive way.