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  "body": "\n## Problem\n\nA solid body moves with constant speed along its axis through a rarefied medium.\nThe particles of the medium do not interact with each other and bounce off the body elastically.\nAmong convex bodies of revolution with base radius $R \u003e 0$ and height $h \u003e 0$, which shape has the least resistance?\n\nIn Newton's model, the body is $\\set{(x, z) \\in \\R^2 \\times \\R : \\abs{x} \\le R, \\ 0 \\le z \\le u(\\abs{x})}$.\nIts profile $u \\colon [0, R] \\to [0, h]$ is [[concave|07Q7CNM5WF]] and nonincreasing.\nThe resistance is proportional to the [[integral functional|JRQZ803Q5Y]]\n$$F(u) = \\int_0^R \\frac{r}{1 + u'(r)^2} \\, dr.$$\nThe problem is to minimise $F$ over such profiles.\n\n## Remarks\n\nNewton stated the answer in the Principia.\nThe optimal profile is not pointed: it has a flat disc of positive radius at the front, followed by a curved part that meets the disc at a nonzero angle.\nThe convexity condition matters.\nWithout it, profiles with steep zigzags make $F(u)$ as small as one likes.\nBut then particles would hit the body more than once, and the model no longer describes the resistance.\nLater work showed that convex bodies without rotational symmetry can have smaller resistance than Newton's body of the same radius and height.\n\n## Explanations\n\n### L1 human\n\nNewton asked how to shape the nose of a body so that it pushes through a thin gas as easily as possible.\nHis answer is surprising: the best nose is blunt, with a flat front, and not a sharp point.\nIt was one of the first problems in which a shape was found by minimising an integral.\n",
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