When a gun fires a bullet at velocity v aboard a plane also traveling at velocity v, the bullet's new velocity can be understood through Newton's Laws of motion and Galilean transformation principles. The bullet's velocity relative to an observer on the ground is the sum of the plane's velocity and the bullet's initial velocity, resulting in a total velocity of 2v. This explanation holds true under the condition that the velocities involved are much less than the speed of light (v << c). The discussion emphasizes the simplicity of this proof using classical mechanics. Understanding these principles is essential for analyzing motion in different reference frames.