2 - MOTION IN A LINEAR RESISTING MEDIUM

Summary

“Even horizontal motion, which if no impediment were offered would be uniform and constant, is altered by the resistance of the air and finally ceases, and here again the less dense the body, the quicker the process.”

Galileo Galilei (1564–1642)

Velocity and Position Vectors

The formulae derived in Chapter 1 relate strictly to a projectile travelling under the influence of constant gravity in a vacuum. When the projectile moves through any fluid medium (gas or liquid) other forces are present due to the slowing-down influence of that medium's particles. The sum of these forces in a direction opposite to the projectile's velocity is called the drag force and for many non-spinning projectiles it is the main effect of the medium. A more detailed discussion of drag will be postponed until the beginning of Chapter 4.

Experiments show that this drag force is usually related in a non-linear way to the velocity of the projectile. When a projectile is moving at moderate or high speeds the non-linear drag force can be approximated by using different powers of the speed over different velocity ranges.

For projectiles moving through air at very low speeds, or for motion through other fluids where the Reynolds number (see Chapter 7) is small, the drag can often be assumed to be directly proportional to the speed. This linear model shows the effect of the inclusion of drag without the mathematics becoming too complicated at first.