BALLISTIC CURVE Fixed friction proportional to speed, increasing initial speed, shooting angle of 60° Ditto with a shooting angle of 45° Ditto with a shooting angle of 30°

 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 - in red, with a resistance proportional to the speed (j(v) = v ) - in green, with a resistance proportional to the square of the speed (j(v) = v² )  © Robert FERRÉOL 2017