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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
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) = )
 


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© Robert FERRÉOL 2017