## Illustration 5.5: Air Friction

In this Illustration we compare the motion of a red projectile launched upward to that of an identical green projectile launched upward but subjected to the force of air friction. To make the motion easier to see, we have given both projectiles a slight horizontal velocity and do not consider frictional effects in this direction either. In addition, we show the free-body diagrams for each projectile (the force of gravity is drawn with a fatter vector so it is easier to see). Restart.

Watch the Position Graph animation and look at the free-body diagram. First, what is the direction of the force of air friction? It opposes motion, just as static and kinetic friction do. Consider the Velocity Graph animation. If we look at the motion on the way up, the velocity is positive, and therefore the force of friction opposes the motion and is downward, hence |ay| > g on the way up. At the top of the arc, the velocity is zero, and hence |ay| = g. On the descent, the velocity is downward, and the force of air friction is therefore upward and hence |ay| < g. Therefore, |ay| is greater on the way up! This is borne out by the Acceleration Graph. At some point, the frictional force has exactly the same size as the force of gravity. When this occurs there is no longer a net force, and the acceleration of the projectile is zero. The velocity corresponding to this situation is called the terminal velocity.

These animations are valid at low speeds. We can experimentally determine that the force of air friction is proportional to the velocity at low speeds, with R = -b v, where R is the resistive or drag force and b is a constant that depends on the properties of the air and the size and shape of the object. One benefit of this model is that the mathematics is a little easier to handle than for the high-speed case.

For massive small objects at high speeds (not depicted, but you can look at Exploration 5.6 to view this model) we can experimentally determine that the force of air friction is proportional to the velocity squared. The magnitude of the drag force can be represented as R = 1/2 Dρ Av2, where ρ is the density of air (mass/volume), A is the cross sectional area of the object, v is the magnitude of the velocity, and D is the drag coefficient (0.2–2.0). Sometimes the drag force is written as bv2 with the assignment that b = 1/2DρA. We can solve for the velocity as a function of time, but it is harder. We must be careful in this model if we have two-dimensional motion, since the x and y motions are no longer independent.

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