# Projectile Envelope - Alternative Derivation?

(see Desmos implementation here)

Consider a projectile $$P$$ launched from the origin at launch velocity $$v$$ and launch angle $$\alpha$$. It can be shown easily that the envelope of projectile trajectories for different values of $$\alpha$$ is given by $$y=h-\frac {x^2}{4h}$$ where $$h=\frac {v^2}{2g}$$. The envelope is a parabola with axes intercepts $$(\pm 2h, 0)$$ and $$(0,h)$$.

Consider a projectile $$Q$$ launched from $$(-2h, 0)$$ at launch velocity $$V=v\sqrt2$$ and launch angle $$\frac {\pi}4$$ concurrently with projectile $$P$$. It can be easily shown that the trajectory of this projectile is the same as the envelope of projectile $$P$$.

Without first working out the algebra, can we come to the conclusions above, and in particular, that this requires $$V=v\sqrt2$$?

Also, projectile $$Q$$ reaches $$(h,0)$$ at the same time as projectile $$P^*$$ (projectile $$P$$ with launch angle $$\frac {\pi}2$$), and $$(2h,0)$$ at the same time when projectile $$P^*$$ reaches $$(0,0)$$ again.

Can this fact be used to conclude that projectile $$Q$$ (with specified parameters) traces out the envelope for projectile $$P$$?