The use of negative numbers in most of today's calculations is natural. But how did the use of negative numbers began in physics? What physical quantity required the introduction of negative numbers and why?
Ancient Greeks painstakingly avoided negative numbers, although they could have come handy in astronomical calculations and number theory, among other places. Brahmagupta in Correctly Established Doctrine of Brahma (c. 630 AD) uses the language of "fortunes" and "debts", which suggests the merchant origin of the negative number concept, but that remains a speculation. Bhaskara II casually mentions negative sides when solving triangles in Lîlâvatî (c. 1150):"[The result] is negative, that is to say, in the contrary direction." Even earlier Liu Hui (c. 260 AD) and later medieval Islamic mathematicians accepted the negatives simply to make the algebra work more generally, see Historically, how did people define multiplication for negative numbers?
As for physics, it is not that any problem "required" them. Unlike many other concepts, where physics was an engine driving mathematical advances, when negative numbers finally made their way into physics it was as an afterthought. Medieval natural philosophers, and even Galileo, did not follow Bhaskara. They show considerable ingenuity in avoiding negatives, in the fashion of ancient Greeks, even where they would have been natural, such as in velocities or temperatures. Oresme (c. 1360 AD), the pioneer of coordinates and (bar) graphs, always adds a constant to his velocity and temperature graphs to avoid negative values. Even Fermat and Descartes still followed suit. Galileo in the Dialogues concerning two new sciences (1638) specifically splits projectile parabolas into two semi-parabolas to avoid having the velocity change sign.
The first physical negatives were, apparently, distances and their derivatives. According to Mumford's What’s so Baffling About Negative Numbers?, we should attribute the first application to Wallis and Newton, who worked independently. Wallis in his Treatise on Algebra (1685) speaks of negative distances traveled:
"As for instance: Supposing a man to have advanced or moved forward (from A to B) 5 yards; and then to retreat (from B to C) 2 yards; If it be asked, how much had he Advanced (upon the whole march) when at C? I find... he has Advanced 3 Yards. But if, having Advanced 5 Yards to B, he thence retreat 8 Yards to D; and it then be asked, How much is he Advanced when at D, or how much Forwarder than when he was at A: I say –3 Yards."
Newton allowed negative numbers for velocities, accelerations, etc., in earlier notes, but did not spell it out in print until after Wallis. In the Universal Arithmetick (1707) he writes:
"Quantities are either Affirmative, or greater than nothing; or Negative, or less than nothing. Thus in human affairs, possessions or stock may be called affirmative goods, and debts negative ones. And so in local motion, progression may be called affirmative motion, and regression negative motion; because the first augments, and the other diminishes the length of the way made."