I found another article to supplement those to which Andrey linked: Eugene I Butikov, "Comment on 'The envelope of projectile trajectories'," Eur. J. Phys. 24 L5-L9, 2003. He also explains the expanding-circles viewpoint that is Andrey's second bullet. He imagines first that that there is no gravity, in which case the particles are on the surface of an expanding sphere whose radius $r$ equals $v t$. "With gravity, this uniformly expanding sphere is falling freely as a whole with the acceleration of free fall." He then finds the envelope of these falling, expanding circles.
Later in the note he considers water drops spun off a spinning, wet bicycle wheel. To continue the 4th-of-July theme, these could be sparks from a spinning sparkler wheel. He proves that again, the envelope of the drops/sparks is a parabola.
