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Exercises

and that give the body’s coordinates as functions of time serve as parametric equations for the body’s motion and path. With vector notation, we can condense these into a single equation that gives the body’s position as a vector function of time. For an object moving in the xy-plane, the component function is zero for all time (that is, identically zero). In this chapter, we  use calculus to study the paths, velocities, and accelerations of moving bodies. As we go along, we will see how our work answers the standard questions about the paths and motions of projectiles, planets, and satellites. In the final section, we

use our new vector calculus to derive Kepler’s laws of planetary motion from Newton’s laws of motion and gravitation. Because the derivatives of vector functions may be computed component by component,

the rules for differentiating vector functions have the same form as the rules for differentiating scalar functions  When we shoot a projectile into the air we usually want to know beforehand how far it will

go (will it reach the target?), how high it will rise (will it clear the hill?), and when it will land (when do we get results?). We get this information from the direction and magnitude

of the projectile’s initial velocity vector, using Newton’s second law of motion   and we will studyThe Vector and Parametric Equations for Ideal Projectile MotionTo derive equations for projectile motion, we assume that the projectile behaves like a particle moving in a vertical coordinate plane and that the only force acting on the projectile

during its flight is the constant force of gravity, which always points straight down. In practice, none of these assumptions really holds. The ground moves beneath the projectile as the earth turns, the air creates a frictional force that varies with the projectile’s speed

and altitude, and the force of gravity changes as the projectile moves along. All this must be taken into account by applying corrections to the predictions of the ideal equations we are about to derive. The corrections, however, are not the subject of this section. Imagine the motions you might experience traveling at high speeds along a path through the air or space. Specifically, imagine the motions of turning to your left or right and the up-and-down motions tending to lift you from, or pin you down to, your seat. Pilots flying through the atmosphere, turning and twisting in flight acrobatics, certainly experience

these motions. Turns that are too tight, descents or climbs that are too steep, or either one coupled with high and increasing speed can cause an aircraft to spin out of control, possibly even to break up in midair, and crash to Earth. In this and the next two sections, we study the features of a curve’s shape that describe mathematically the sharpness of its turning and its twisting perpendicular to the forward motion