Curve studied by Varignon in 1699.

An isochronous curve of Varignon is a curve such that if a particle descends along it by the pull of gravity, then the vertical component of the speed is constant, when the gravitational force is supposed to have constant norm and to be oriented in the direction of the centre of the Earth (which is physically false; did Varignon know the laws of gravitation of Newton in 1699?).
 

With O as the centre of the Earth, and a starting point at  with a zero horizontal speed, the conservation of the energy gives , i.e.  since . Hence the kinematic equations of motion:  which can be integrated to: .

See also the isochronous curve of Leibniz.
 

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© Robert FERRÉOL  2017