When I studied physics in high school, we touched on ballistics by calculating the path of a projectile, ignoring air resistance.
I knew that taking air resistance into account would complicate things, but it turns out that drag — which is proportional to velocity squared, in the opposite direction — makes the calculus intractable. Ballistics tables are computed by brute force.
Now an Indian-born “German” teenager, Shouryya Ray, has (kind of, sort of) solved the 300-year-old mathematical riddle posed by Sir Isaac Newton:
Ray won a research award for his efforts and has been labeled a genius by the German media, but he put it down to “curiosity and schoolboy naivety.”
“When it was explained to us that the problems had no solutions, I thought to myself, ‘well, there’s no harm in trying,’” he said.
Ray’s family moved to Germany when he was 12 after his engineer father got a job at a technical college. He said his father instilled in him a “hunger for mathematics” and taught him calculus at the age of six.
Ray’s father, Subhashis, said his son’s mathematical prowess quickly outstripped his own considerable knowledge.
“He never discussed his project with me before it was finished and the mathematics he used are far beyond my reach,” he said.
Despite not speaking a word of German when he arrived, Ray will this week sit Germany’s high school leaving exams, two years ahead of his peers.
This solution is implicit, therefore has little use in actual calculations as you would need to numerically solve it in order to use it, might as well solve the differential equation numerically directly. Exact solutions similar to the one presented here have been known since 1977 in a paper I posted in another thread . Anyway the trick used to solve the ODE is quite clever, especially for a 16 year old.