Ballistic curve

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 = 0, we get the parabola.

2) Case where = 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 ( = v )
- in green, with a resistance proportional to the square of the speed ( = v² )

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