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.

  • $\begingroup$ Somewhat off-topic: there is a neat online game where one can get some practice with parabolic trajectories :) Or, are they in fact elliptical? onlinegames.com/basketball $\endgroup$ Jul 4, 2010 at 15:27

3 Answers 3

  • E. Torricelli, who was the last Galileo's secretary, suggested a purely geometrical method to find the envelope in his De motu Proiectorum. He also coined the term `parabola of safety'. Apparently it was the first example of computation of an envelope. The method is briefly described in this note.

  • Another approach is to launch identical missiles with the same velocity at all possible angles simultaneously. At time $t$, their positions describe a circle $$x^2+\left(y-\frac{t^2}{2}\right)^2=(vt)^2.$$ The latter equation has a unique solution in $t$ provided $(x,y)$ belongs to the parabola $$y=\frac{v^2}{2}-\frac{x^2}{2v^2}.$$

  • In the case of missiles moving in a Kepler field (with the attractive potential $\sim -1/r$), the envelope of elliptic trajectories is indeed an ellipse. A web search gave the nice short article which contains several elementary geometric proofs of this and related results.

Edit. A free version of J.-M. Richard article can be found here.

  • $\begingroup$ The first links seems to be broken and the last isn't open-access, unfortunately; the second point is quite cool though. $\endgroup$ Jul 3, 2010 at 19:37
  • $\begingroup$ Daniel, thank you for the comment. I've added an arxiv link to the second article. As for the first one, the link is jstor.org/stable/3620177?seq=1 $\endgroup$ Jul 3, 2010 at 20:12
  • $\begingroup$ Anyway, it is probably a MO bug, as both links work in my web browser. $\endgroup$ Jul 3, 2010 at 20:14
  • $\begingroup$ @Andrey: Thanks so much!! It is cool that this question has such deep historical roots. I cannot access the JSTOR papers (must be a JSTOR permissions issue?), but I can access the arXiv paper. The freely falling expanding sphere is a beautiful viewpoint! And that the envelope of the elliptical trajectories is also an ellipse is quite satisfying. I am still hopeful there is a nested/tangent cones viewpoint, perhaps extractable from Richard's article. Will ponder this as I watch the fireworks... :-) $\endgroup$ Jul 3, 2010 at 23:11
  • $\begingroup$ @Joseph: Thank you for the comment. Unfortunately, I have not found a free version of the JSTOR paper. $\endgroup$ Jul 4, 2010 at 12:18

It is easy to see that all these parabolas have the same directrix. Height of a directirix correspond to energy of the body. So you have the family of parabolas with the common point $P$ and the directrix $l$. It is easy to prove, (using just definition of parabola as a locus of points...) that all of the touched the parabola with the focus at $P$ and the directrix $l_1$, which parallel $l$ (actually $l$ is midline of $P$ and $l_1$).

The same holds for sun-earth set. If Earth decides to fly in other direction (but with the same speed) its path will be always touch the fixed ellipse with foci in Sun and this position of Earth.


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.

  • $\begingroup$ This is interesting. So it's not only mathematicians who love to go back and and rederive classical results. Some physicists seem to have this `bug' too. $\endgroup$ Jul 4, 2010 at 14:59

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