Calculus: Analytic Geometry and Calculus, with Vectors

46 Analytic geometry in two dimensions

The physical significance of the constants in (1.671), (1.672), and

(1.673) is worthy of notice. As we see by putting t = 0 in (1.671), so is

the value of s (the displacement) when t = 0, so so is called the initial

displacement. As we see by putting t = 0 in (1.672), vo is the scalar

velocity when t = 0, so vo is called the initial scalar velocity. It is easily

seen from (1.672) that v = 0 when t = -vo/g. The values (if any) of t

for which s = 0 can be obtained by putting s = 0 in (1.671) and solving

the resulting quadratic equation for t. In many applications, the space

and time coordinates are so chosen that the initial displacement so and

initial velocity vo are both 0. In this case (1.671) reduces to the simpler

formula

(1.674)

The related formulas

(1.675)

S = 1 gt2.

g

which give the time required for the body to fall a distance s and the speed

attained when the body has fallen a distances, are often useful.

We conclude with a remark about uniform circular motion. Suppose a

particle starts at time t = 0 on the positive x axis and moves, with angular

speed w (omega) radians per second, in the positive (counterclockwise)

direction around the circle of radius R having its center at the origin.

Letting r denote the vector running from the origin to the particle P

at time t gives the first of the formulas

(1.681) r = R( cos wti + sin wtj)

(1.682) v = wR(-sin wti + cos wtj)

(1.683) a = -w2R( cos wti + sin wtj)

where, as in Figure 1.684, i and j are unit vectors having the directions of

Figure 1.684

the positive x and y axes. Application of rules of Chapter 3 then gives

(1.682) and (1.683) as rapidly as we can write them. Looking at (1.681)

and (1.683) shows that a = -w2r and hence that P is always accelerated