UNIT 8
VELOCITY, ACCELERATION AND PROJECTILES
INTRO: In this Unit we will study velocity and acceleration along trajectories, and, in partic
ular, projectile problems.
1. Vector Functions
In Units 1 and 2 we studied trajectories given by parametric equations,
~
r
(
t
) =
< x
(
t
)
, y
(
t
)
, z
(
t
)
> .
Since
~
r
(
t
) is a function of
t
which gives a vector at each
t
value, namely
< x
(
t
)
, y
(
t
)
, z
(
t
)
>
, we
call it a
vector function
, or more accurately, a
vectorvalued function
. The differentiation and
integration of such vectorvalued functions is defined and carried out componentwise, so that
d
dt
< x
(
t
)
, y
(
t
)
, z
(
t
)
>
=
< x
0
(
t
)
, y
0
(
t
)
, z
0
(
t
)
>
and
Z
< x
(
t
)
, y
(
t
)
, z
(
t
)
> dt
=
<
Z
x
(
t
)
dt
+
c
1
,
Z
y
(
t
)
dt
+
c
2
,
Z
z
(
t
)
dt
+
c
3
> .
Note that the indefinite integral of a vectorvalued function introduces a different arbitrary
constant in each component. One may check that this is valid by differentiating the result, in
which case all of the constants disappear, as they should.
Because the same trajectory can be parametrized many ways, there is not, in general, any
physical significance to the derivative, or rather, to its magnitude. That is to say, the equations
~
q
(
t
) =
< t, t
2
+ 1
,
2
t
3
>
and
~
r
(
t
) =
< t
3
, t
6
+ 1
,
2
t
9
>
describe precisely the same trajectory, but
their derivatives
~
q
0
(
t
) =
<
1
,
2
t,
6
t
2
>
and
~
r
0
(
t
) =
<
3
t
2
,
6
t
5
,
18
t
8
>
are quite different. However,
when the parameter t is the time,
then the derivatives of the trajectory
~
r
(
t
) give the velocity
~v
(
t
) and the acceleration
~a
(
t
) of an object traveling along the curve.
Note on Notation
: When
t
is just a parameter, then writing the parametric equations
~
F
(
t
) =
<
f
(
t
)
, g
(
t
)
, h
(
t
)
>
is not a problem. However, when the parametric equations are intended to
describe a physical trajectory in the xy plane or in 3space, and
t
is the time, then parametric
equations for the trajectory such as
~
r
(
t
) =
< t,
sin
t >
have a problem with units. In fact, if
~
r
(
t
)
is expected to give the location of an object in the xy plane at time
t
, then the components
of
~
r
(
t
) must have units of length, say feet. But if
~
r
(
t
) =
< t,
sin
t >
and time is in seconds, the
first component of
~
r
will have units of seconds, and the second component will make even less
sense, since the sine function is unitless, and, moreover, the argument of a trig function must
be in radians or degrees, certainly not seconds of time. To get around this, an engineering or
science application will generally write this same trajectory as
~
r
(
t
) =
< at, b
sin
ωt >
and then
specify that
a
is the constant 1 ft/sec,
b
is the constant 1 ft, and
ω
is the constant 1 rad/sec, or
1 sec

1
. For pedagogical reasons, we prefer not to have all these constants in our mathematics
VECTOR GEOMETRY
96
GREENBERG