instance, it will be quicker to approach the solution via an integrating factor . (2c) (2d) (2e) (2f) (2g) And by integration we find: (3) Solving for our initial conditions: (2h) (3a) With a bit of algebra to simplify (3a): (3b) The total time of the journey in the presence of air resistance (more specifically, when ) can be calculated by the same strategy as above, namely, we solve the equation . While in the case of zero air resistance this equation can be solved elementarily, here we shall need the Lambert W function. The equation is of the form , and such an equation can be transformed into an equation solvable by the function (see an example of such a transformation here). Some algebra shows that the total time of flight, in closed form, is given as . Trajectory of a projectile with Newton drag The most typical case of air resistance, for the case of Reynolds numbers above about 1000 is Newton drag with a drag force proportional to the speed squared, . In air, which has a kinematic viscosity around , this means that the product of speed and diameter must be more than about . Unfortunately, the equations of motion can not be easily solved analytically for this case. Therefore, a numerical solution will be examined. The following assumptions are made: Constant gravitational acceleration Air resistance is given by the following drag formula, Where: FD is the drag force c is the drag coefficient ρ is the air density A is the cross sectional area of the projectile μ = k/m = cρA/(2m) Special cases Even though the general case of a projectile with Newton drag cannot be solved analytically, some special cases can. Here we denote the terminal velocity in free-fall as and the characteristic settling time constant . Near-horizontal motion: In case the motion is almost horizontal, , such as a flying bullet, the vertical velocity component has very little influence on the horizontal motion. In this case: The same pattern applies for motion with friction along a line in any direction, when gravity is negligible. It also applies when vertical motion is prevented, such as for a moving car with its engine off. Vertical motion upward: Here and where is the initial upward velocity at and the initial position is . A projectile can not rise longer than vertically before it reaches the