the absolute speed: . The speed-dependence of the friction force is linear () at very low speeds (Stokes drag) and quadratic () at larger speeds (Newton drag). The transition between these behaviours is determined by the Reynolds number, which depends on speed, object size and kinematic viscosity of the medium. For Reynolds numbers below about 1000, the dependence is linear, above it becomes quadratic. In air, which has a kinematic viscosity around , this means that the drag force becomes quadratic in v when the product of speed and diameter is more than about , which is typically the case for projectiles. Stokes drag: (for ) Newton drag: (for ) The free body diagram on the right is for a projectile that experiences air resistance and the effects of gravity. Here, air resistance is assumed to be in the direction opposite of the projectile's velocity: Trajectory of a projectile with Stokes drag Stokes drag, where , only applies at very low speed in air, and is thus not the typical case for projectiles. However, the linear dependence of on causes a very simple differential equation of motion in which the two cartesian components become completely independent, and thus easier to solve. Here, , and will be used to denote the initial velocity, the velocity along the direction of x and the velocity along the direction of y, respectively. The mass of the projectile will be denoted by m, and . For the derivation only the case where is considered. Again, the projectile is fired from the origin (0,0). The relationships that represent the motion of the particle are derived by Newton's Second Law, both in the x and y directions. In the x direction and in the y direction . This implies that: (1), and (2) Solving (1) is an elementary differential equation, thus the steps leading to a unique solution for vx and, subsequently, x will not be enumerated. Given the initial conditions (where vx0 is understood to be the x component of the initial velocity) and for : (1a) (1b) While (1) is solved much in the same way, (2) is of distinct interest because of its non-homogeneous nature. Hence, we will be extensively solving (2). Note that in this case the initial conditions are used and when . (2) (2a) This first order, linear, non-homogeneous differential equation may be solved a number of ways; however, in this