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BALLISTIC CURVE
Fixed friction proportional to speed, increasing initial speed, shooting angle of 60° 


Curve studied by Jean Bernoulli in 1719, Euler in 1753, Legendre in 1782 and Jacobi in 1842. 
The ballistic curves are the trajectories of a massive point subject to a uniform gravitational field and a fluid friction force, in the opposite direction of the velocity vector, its intensity being proportional to a certain function j(v) of the absolute velocity.
1) When j(v) = 0, we get the parabola.
2) Case where j(v) = v (experimentally obtained for a low velocity; this resistance is called "viscosity")
(Linear) differential equation of motion:
(h = coefficient of friction, m = mass of the point g gravitational acceleration) Cartesian parametrization for : (initial condition M(0) = 0) with, , and . Cartesian equation: (whereas when there is no friction, it is: with ). Transcendental curve (as opposed to the parabola). 

We get curves which, as opposed to the parabolas, have a vertical asymptote at their right end, and an oblique asymptotic branch without asymptote, at the left end.
Opposite, figure composed of the trajectories starting from a fixed point with constant initial speed, shooting angle of 45°, and an increasing friction coefficient. 
Figure composed of the trajectories starting from a fixed point with given initial speed, along with the envelope of these trajectories (called safety parabola even though it is not a parabola).
Compare to the case without friction. 
Remark: the Cartesian equation of these curves shows that they are the medians of a line and a logarithmic curve, along the asymptote of the logarithmic curve.
3) General case.
Differential equation of motion:
which can be written:
hence , so that, with , : differential equation giving as a function of u. Substitution of into (1) gives hence, using , the parametrization of the curve as a function of u. 
Opposite, comparison between 3 ballistic curves, under the same shooting angle and with the same initial speed:
 in blue, in space

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© Robert FERRÉOL, Jacques MANDONNET 2017