TITLE: Parabolic envelope of fireworks
QUESTION [38 upvotes]: The envelope of parabolic trajectories from a common launch point is itself a parabola.
In the U.S. soon many will have a chance to observe this fact directly, as the 4th of July is traditionally celebrated with fireworks.
If the launch point is the origin, and the trajectory starts off at angle $\theta$ and velocity $v$, then under unit gravity it follows that the parabola
$$
y = x \tan \theta  - [x^2 /(2 v^2)] (1 + \tan^2 \theta)
$$
and the envelope of all such trajectories, is another parabola:
$$
y = v^2 /2 - x^2 / (2v^2)
$$

                


These equations are not difficult to derive.
I have two questions.
First, is there a way to see that the envelope of parabolic trajectories is itself a parabola, without computing these equations?
Is there a purely geometric argument?
Perhaps there is a way to nest cones and obtain the above picture through conic sections, but I couldn't see it.
Second, of course the trajectories are actually pieces of ellipses, not parabolas, if we follow the true inverse-square law of gravity.
Is the envelope of these elliptical trajectories also an ellipse?
(I didn't try to work out the equations.)
Perhaps the same geometric viewpoint (if it exists) could apply, e.g. by slightly tilting the sections.

REPLY [3 votes]: 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 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.