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Problem

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problem

When discussing the orbits of three stars of equal mass under the influence of each other's gravity alone, intuition might suggest a co-circular orbit. Recent research has found that the motion of three stars of equal mass under the influence of each other's gravity can also present an eight-shaped orbit. It is assumed that the position and shape of the orbit does not change over time.

\\(O\\) and \\(P\\) are points on the eight-shaped orbit. Let \\(O\\) be at the centre of the eight-shaped orbit (where the curve intersects itself). Let \\(P\\) be at one of the points on the orbit furthest from \\(O\\).
We enumerate the stars with \\(1\\), \\(2\\) and \\(3\\) in the order they pass a given point. At a given moment when star \\(1\\) passes point \\(O\\), let \\(O_2\\) and \\(O_3\\) denote the positions of stars \\(2\\) and \\(3\\) respectively at this moment. Likewise, at a given moment when star \\(1\\) passes through point \\(P\\), let \\(P_2\\) and \\(P_3\\) denote the positions of stars \\(2\\) and \\(3\\).\
The distances \\(OO_2 = 145.2 L\\), and \\(PP_2 = 242 L\\) are known, as well as the angle \\(\\angle OPP_2 = 11.6^\\circ\\). Here, \\(L\\) is an arbitrary unit of length.

Find a numerical value for the ratio of the distances \\(s = \\frac{OO_2}{OP_2} \\), rounded to 3 significant figures.

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subject

physics

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Source and history

upstream_id / Original problem ID
2c46387b-36e6-43cb-9e73-dbf1341bfafd
task_id
bd9eb68c-9403-5c93-a243-0b9ea1d015d6
task_key
olympiad--test--2c46387b~2d36e6~2d43cb~2d9e73~2ddbf1341bfafd
task_revision_id
4
config / split
olympiad / test

Official source

initial import