3 MOTION IN 3 DIMENSIONS 3.8 Vector velocity and vector acceleration

By analogy with the 1 - dimensional equation (2.3), the body’s vector velocity v =

(vx, vy, vz) is simply the derivative of r with respect to t. In other words,

v(t) = lim r(t^ +^ ∆t)^ −^ r(t)^

dr

∆t (^0) ∆t =^ dt.^ (3.15)^
When written in component form
→
, the above definition yields
dx
vx =
vy =
, (3.16)
dt
dy
, (3.17)
dt
dz
vz =.^ (3.18)^
dt
Thus, the x-component of velocity is simply the time derivative of the x-coordinate,
and so on.
By analogy with the 1 - dimensional equation (2.6), the body’s vector acceler-
ation a = (ax, ay, az) is simply the derivative of v with respect to t. In other
words,
a(t) = lim v(t^ +^ ∆t)^ −^ v(t)^ dv^ d
(^2) r
∆t (^0) ∆t =^ dt =^ dt^2.^ (3.19)^
When written in component
→
form, the above definition yields
ax =
ay =
az =
dvx

dt
dvy

dt
dvz

dt
d^2 x
, (3.20)
dt^2
d^2 y
, (3.21)
dt^2
d^2 z

. (3.22)
dt^2

Thus, the x-component of acceleration is simply the time derivative of the x-
component of velocity, and so on.

As an example, suppose that the coordinates of the body are given by

x = sin t, (3.23)^

y = cos t, (3.24)^

z = 3 t. (3.25)