Calculus: Analytic Geometry and Calculus, with Vectors

3.7 Rates, velocities

11 This problem involves the uniform circular helical
motion of a particle Q in E3 which runs up the helix (spiral
staircase) of Figure 3.792 in such a way that its projection
P upon the xy plane executes the uniform circular motion
of Problem 9 whiles its z coordinate increases at the posi-
tive rate b units per second. Supposing that Q occupies
the position (a,0,0) when t = 0 and letting r = OQ, show
that

r = a(cos wti + sin wtj) + btk

v = aw(-sin wti + cos wtj) + bk

a = -aw2(cos wti + sin wtj).

183

Figure 3.792

Find the speed of Q. Remark: One who gets interested in this helix may try

to find the length of one turn by two different methods. First, use the speed of Q

in an appropriate way. Second, find out what happens when the cylinder upon

which the helix lies is cut along a vertical generator and rolled out flat.

12 A projectile P moves in such a way that its displacement vector at time

t is

(1) r = (vo cos a)ti + [(vo sin a)t - .gt2]j,

where a, vo, g are constants for which 0 < a < 7r/2, vo > 0, g > 0. Show that

its velocity at time t is

(2) v = [(vo cos a)i + (vo sin a)j] - gtJ.

Show that a = -gj. Show that the coordinates x, y of P at time t are

(3) x = (vo cos a)t, y = (vo sin a)t - gt2.

Eliminate t to obtain the equation

gx2

(4) y = (tan a)x -^2

C 2vo cost a

and note that the path of the projectile is a part of a parabola. Show that

y = 0 when t = 0 and that the projectile is then at the origin. Show that y = 0

when i = (2vo sin a)/g and that the projectile is then at the point (R,0), where

(5)

R 2v2 o sin a cos a=vo sin 2a

g g

This number R is called the range of the projectile, and this range is clearly a

maximum when sin 2a = 1 and hence when 2a = a/2 and a = 7r/4. Show that

the initial velocity (velocity at time t = 0) is

(6) (vo cos a)i + (vo sin a)j

and that this makes the angle a with the positive x axis. Show that the initial

speed is vo. Tell, in terms of vectors, how the velocity at later times is related

to the initial velocity. Find the velocity (not merely speed) of the projectile

when it hits the point (R,0).