Slick Projectiles

The slick solution is based on the fact that the point product of the velocity vectors of projectiles is zero when they are penpendicular one to the other:

$\vec{V}_1 • \vec{V}_2=-v_1 v_2+2gh=0 \rightarrow h=\frac{v_1v_2}{2g}$

The first term on the right side of the first equation is negative because the horzontal componenent of the velocities are antiparallel, both projectiles fell the same lenght, thus, they have the same velocity's vertical component. The rest are simple kinematics.

The relative movement of a projectile with respecto to the other is:

$x_r=(v_1+v_2)t ; y_r=0$ ("r" stands for relative)

The projectiles' vertical position depend on time by:

$y=\frac{gt^2}{2} \rightarrow t=\sqrt{\frac{2y}{g}}$

So:

$x_r=(v_1+v_2) \sqrt{\frac{2y}{g}}$

For y=h (when they are perpendicular):

$x_r=(v_1+v_2) \frac{\sqrt{v_1v_2}}{g} =1m$

The "1m" is obtained after replacing values.

Below is a diagram on how it would look like when the velocities are perpendicular where $θ_1=tan^{-1}(\frac{gt}{1}$ ) and $θ_2=tan^{-1}(\frac{gt}{4}$ ). Since we know that $θ_1+θ_2=90^o$ , we can solve for $t$ which gives $t=0.2s$ . Horizontal separation $=(4+1)t=1m$ .