The air resistance that an object encounters is proportional to its cross-sectional area and to its velocity. Notice, that like friction, air resistance opposes motion. That is, if the object is rising through the air, air resistance acts downward; if it is falling, air resistance acts upwards. Notice that in the accompanying diagram, that the air resistance vectors vary in length. This signifies that its magnitude changes as the projectile's velocity changes.

 

Notice that while rising, both air resistance and weight oppose the object's upward motion. This results in the projectile losing speed at a rate greater than 9.8 m/sec² and subsequently, rising to a lower apex height. On the way down, air resistance increases as the object gains velocity. But since air resistance and weight oppose each other, the projectile is gaining speed at an ever decreasing rate; that is, its acceleration is decreasing. Eventually, when air resistance and weight become equal, the projectile's downward acceleration ceases and the object reaches a state of dynamic equilibrium called terminal velocity. The remainder of its trajectory is traveled at this constant speed. All of these factors lead to a projectile spending more time falling than it does rising when in the presence of air resistance. The symmetry of freefall is lost for a projectile thrown up into the air.

 

The air resistance, or drag force, is generally expressed as AR = F_drag = -kvⁿ where k and n are constants that depend on the geometry of the object and the medium through which it is moving. When terminal speed is reached,

 

net F = 0
AR + mg = 0
-kvⁿ + mg = 0
-kvⁿ = -mg
kvⁿ = mg
v = (mg/k)^1/n

 

Obviously, the larger the values of k and n, the smaller the object's terminal speed.