Newton’s second law gives SF x 52Dvv x 5 ma x SF y 52mg 2 Dvv y 5 ma y and the components of acceleration including the effects of both gravity and air drag are ax 521D/m 2 vvx ay 52g 2 1 D/m 2 vvy (T1.2) The constant D depends on the density r of air, the silhouette area A of the body (its area as seen from the front), and a dimensionless constant C called the drag coefficient that depends on the shape of the body. The direction of f is opposite the direction of µ, so we can write fù–Dvµ and the components of f are 2 f x 52Dvv x Note that each component is opposite in direction to the corresponding compo- 2 2 nent of velocity and that f 5 'fx 1 fy. We’ll assume that the air is still, so µ is the velocity of the projectile relative to the ground as well as to the air (Fig. At the speed of a tossed tennis ball or faster, the magnitude f of the air drag force is approximately proportional to the square of the projectile’s speed relative to the air: (T1.1) where v 2 f 5 Dv 2 2 = vx + vy. We must now include the air drag force acting on the projectile. The components of the projectile’s acceleration were simply a x 5 0 a y 52g The +x-axis is horizontal, and the +y-axis is vertically upward. When we omitted air drag, the only force acting on a projectile with mass m was its weight wùmg.

Fortunately, it is fairly easy to make quite precise numerical approximations to these solutions, using a computer. It’s not difficult to include the force of air resistance in the equations for a projectile, but solving them for the position and velocity as functions of time, or the shape of the path, can get quite complex. We’d now like to extend this analysis to a projectile moving in a plane. In Section 5.3 we considered how a fluid resistance force affected a body falling straight down.

But in fact air resistance (often called air drag, or simply drag) has a major effect on the motion of many objects, including tennis balls, bicycle riders, and airplanes. Topic 1 | Projectile Motion with Air Resistance A Case Study in Computer Analysis In our study of projectile motion, we assumed that air-resistance effects are negligibly small.

Projectile motion with air resistance