Yaron has asked because that a "physical" explanation, but there is a "mathematical" one the is important enough that every physicist should be acquainted with. I placed scare quotes about those words, since the distinction is no as clear together some people would favor it to be. In this case the core "mathematical" reality is predicated top top a aramuseum.org assumption around the nature of motion.

You are watching: Why do complementary angles have the same range

Let"s start by establishing the behavior at a pair of vital special case, and in both instances I shall i think that we throw the ball from zero height, and I shall specify the selection as the horizontal displacement indigenous the point of relax of the very first point after relax at i m sorry the ball has height zero.1

**Horizontal throw:** ($ heta = 0^circ$) The sphere starts at height zero and is released relocating horizontally and also so it automatically at height zero. Variety is 0.

**Vertical throw:** ($ heta = 90^circ$) The sphere goes right up and also falls straight earlier down landing in ~ the point from i beg your pardon it was launched. Variety is 0.

**Maximum range** ($ heta = heta_max$) We expropriate from endure that over there is part angle in ~ which we get the maximum range.

Now us add an essential physical fact: the motion of the sphere is (mathematically) continuous. From this we conclude the the variety as a duty of angle have to be a constant function.2

At that allude we can pick any selection achieved at short angle ($0^circ

These type of debates (relying top top the continuity of various physical quantities under real conditions) are advantageous for creating expectations for many simple systems.

1 I"m likewise assuming no wind (but not necessarily no atmosphere3) and no inertial-pseudo pressures that we need care about.

2 ns think the https://math.aramuseum.org.com/a/430341/8422 consist of the case of integrating the equations the motion.

See more: The Loss Of An Electron By A Molecule Is Called ? Redox Reactions

3 The closed type analysis displayed in several of the various other answers is dependence on a zero wait resistance condition, but this argument works fine v air resistance. It will even work in a continuous breeze, yet finding the zero-range borders is then non-trivial.