Problem $4$:$$ Consider a projectile that lands at the same height from which it was launched.

Show that the speed with which the projectile strikes the ground is independent of launch angle. You do not need to enter anything for this part.

The following trig identity may be helpful for the questions below: sin$(2$$)=2$sin$($$)$cos$($$)$sin$(2$$)=2$sin$($$)$cos$($$)$

Part $($a$)$ If is the launch angle, measured from the horizontal, determine the value of that maximizes the horizontal range of the projectile.