When I search for the history of projectile motion, I mostly find outlines starting with Aristotle and discussing Ibn Sina, Tartaglia, Galileo, Newton, etc. and perhaps with a few more Europeans in the gaps.
Ibn Sina wasn't Western, but he was expanding on Aristotle's work.
Were there models of projectile motion essentially independent of that line of reasoning, for example Chinese or Indian models, in the time before Galileo's work spread to those places?
Needham's Science and Civilisation in China series has a few volumes mentioning ballistics.
Volume 3, Mathematics and the Sciences of the Heavens and Earth, has this short passage on the absence of proper ballistics science in ancient China (p. 167):
The Chinese should have been interested in mechanics for ships, in hydrostatics for their vast canal system (like the Dutch), in ballistics for guns (after all, they had possessed gunpowderf four centuries before Europe), and in pumps for mines. If they were not, could not the answer be sought in the fact that little or no private profit was to be gained from any of these things in Chinese society, dominated by its imperial bureaucracy?
Volume 4, Physics and Physical Technology, part 1, Physics, has this highly speculative passage about the Mozi (p. 58):
And now we find something at any rate extremely like it in the Mo Ching of the -4th or - 3rd century (Cs 49), where motion is said to be due to the absence of an opposing force. The Mohist technical term 'supporting pillar' is to be understood only as that force which in Newton's first law changes the otherwise permanent state of motion of the moving body (Cs 50). The exposition distinctly states that if there is no such force the motion will never stop. Its writer seems to be trying, too, to describe non-linear or deflected motion as 'motion which is not quite motion in the fullest sense'. What remains in these brief fragments is so striking that we may be allowed to believe that if more of the physics of the Mohist school had been preserved, we should have found in it some discussion of trajectories, the effect of gravity, and so on. If the Mohists had no technical term corresponding to impetus, at least they did not suffer from the concept of 'natural place' or the awkward idea of antiperistasis.
But the more interesting passage is perhaps to be found in Volume 5, Chemistry and Chemical Technology, part 7, Military Technology: The Gunpowder Epic, pp. 390$-$391:
As is generally known, the earliest Western speculations about the path of a projectile supposed it to move in a straight line for a while, before succumbing finally to the influence of gravity and then falling downwards in an equally straight line, not unlike the course of the mortar shell seen in Fig. 148. This was the conception in the days of Nicolo Tartaglia (+1537, +1546).c But Galileo (+1638) and Torricelli (+1644) proposed a parabolic trajectory,d and this eventually became more like a hyperbola when the resistance of the air was fully taken into account, as by Newton (+1674) and later mathematicians.e
In East Asia ballistics was pursued more in Japan than in China, but the connections were close. The famous Inatomi family of gunsmithsf left many MS books still extant on the theory and practice of gunnery, notably one of +1607 to +1610 in twenty-nine large volumes.g The most outstanding member of the family, Inatomi Naoie, recorded a tradition that Sasaki Shyo-huziro had first learnt the art in China, and then transmitted it to his grandfather Inatomi Sagami-no-kami Naotoki; this would take us back to +1500 or earlier, certainly before the arrival of either the Turkish or the Portuguese musket (cf. pp. 440 ff. below) in China and Japan, and would suggest that the Chinese hand-guns of the +15th century, probably with serpentines,a had begun the affair.b By +1618 trajectories were being studied by Shimizu Hidemasa, who visualised a slow rise followed by a slow fall.c Then from +1659 onwardsd the parabolic trajectory was proposed, first in the Kaisan-ki1 (Book of Improved Mathematics) by Yamada Shigemasa2,e then in the remarkable work of Nozawa Sadanagra3, the San Kyukai4 (Mathematics in Nine Chapters) of +1677.f This book, which accompanied the illustration of the curve with complicated quadratic equations, was the first Japanese treatise to explain physical phenomena using mathematical formulae. There may have been some Jesuit or other Western influence here,g but Nozawa's view of the world was at least as much Chinese, based on the Yin-Yang theory, decimal metrology,h and the standard pitch-pipe dimensions.i The Suan Fa Thung Tsung5 (Systematic Treatise on Arithmetic) of Chhêng Ta-Wei6 (+1592)j had been translated into Japanese only two years before Nozawa's own writing and he was probably strongly influenced by it. Lastly there was the extension of Yamada's work by Mochinaga Toyotsugu7 & Ohashi Takusei8, the Kaisan-ki Komoku9, which continued to speak of gunshot parabolas.
So, it appears that Japanese scholars had developed some kind of ballistics in the early seventeenth century, but it is hard to tell if this was achieved completely outside from Western influence... I can't copy all the footnotes here, but Needham notes that some of this happened after the closure of the country to foreign influence, when Christian religion had been outlawed, Latin Jesuits expelled, and way before the rangaku (Dutch learning) period had begun.
Yes, but only intuitive ones that every saw as obvious and hence requiring as much comment as noticing that water was wet or that dropping a stone describes a straight line.
The essential characteristics of projectile motion is that it traces an arc. This is obvious from our knowledge of archery and spears. And this obviously goes back a long time - into prehistory.
