Let us adopt a Cartesian coordinate system whose origin coincides with the launch point, and whose -axis points vertically upward. Let the initial velocity of the projectile lie in the - plane. Note that, since neither gravity nor the drag force cause the projectile to move out of the - plane, we can effectively ignore the coordinate in this problem.

The equation of motion of our projectile is written

(175) |

Here, is the

Integrating Equation (176), we obtain

(178) |

(179) |

It is clear, from the above equation, that air drag causes the projectile's horizontal velocity, which would otherwise be constant, to

Integrating Equation (177), we get

(181) |

(182) |

It thus follows, from Equations (180) and (183), that if the projectile stays in the air much longer than a time of order then it ends up falling vertically downward at the terminal velocity, , irrespective of its initial launch angle.

Integration of (180) yields

(184) |

which is the standard result in the absence of air drag. In the opposite limit, , we get

The above expression clearly sets an effective upper limit on how far the projectile can travel in the horizontal direction.

Integration of (183) gives

(187) |

(188) |

(189) |

It is clear, from the previous two equations, that the time
of flight of the projectile (*i.e.*, the time at which , excluding the
trivial result ) is

(190) |

(191) |

when , and

when . Equation (192) is, of course, the standard result without air resistance. This result implies that, in the absence of air resistance, the maximum horizontal range, , is achieved when the launch angle takes the value . On the other hand, Equation (193) implies that, in the presence of air resistance, the maximum horizontal range, , is achieved when is made as small as possible. However, cannot be made too small, since expression (193) is only valid when . In fact, assuming that , the maximum horizontal range, , is achieved when . We thus conclude that if air resistance is significant then it causes the horizontal range of the projectile to scale

Figure 11 shows some example trajectories calculated, from the above model, with the same launch
angle, , but with different values of the ratio . Here,
and
. The solid, short-dashed,
long-dashed, and dot-dashed curves correspond to , , ,
and , respectively. It can be seen that as the air resistance strength
increases (*i.e.*, as increases), the range of the
projectile decreases. Furthermore, there is always an initial time interval
during which the trajectory is identical to that calculated in the absence
of air resistance (*i.e.*, ). Finally, in the presence of
air resistance, the projectile tends to fall more steeply than it rises.
Indeed, in the presence of strong air resistance (*i.e.*, ), the projectile falls almost
vertically.