The same question twice: gravity that pulls straight down, and gravity that pulls toward a point.
Under downward gravity, 45° goes furthest. Under point gravity there is no furthest — you either fall back or leave.
Change the launch, then switch the law of gravity.
Both halves of this are the same exercise: you are told the acceleration, and you integrate twice to get the path. What changes is only what the acceleration vector does.
Make it a constant pointing down and the answer is a parabola, every time. The horizontal motion has no force on it at all so it just ticks along at constant speed, while the vertical motion is the one-variable falling-body problem. Splitting the vector equation into components is what makes it easy — and 45° maximizing the range drops out of the algebra.
Now point the acceleration at a fixed centre instead, with strength falling off as 1/r², and the parabolas become ellipses. The same second-order equation, a different right-hand side, and suddenly you have planetary orbits.
The shaded wedges are Kepler's second law: the radius sweeps equal areas in equal times, so the body races through close approach and dawdles at the far end. Each wedge covers the same slice of time, and you can see them stretch thin and long near the centre. It follows from nothing but the force pointing at the centre — no inverse square needed — because that is exactly the condition for angular momentum to be conserved, and swept area is angular momentum in disguise.