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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 $P$?

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