Parametric Equations Examples
Example 23: Use of Parametric Equations 

A particle moves along a path described by the vector equation
with
R given in meters and
time in seconds. 

(a) Eliminate time t from the parametric equations
and
to
obtain the path equation:
(b) Plot the path. Note that though the path shows both an upper and lower leg, the particle only moves along the upper leg. 

(c)
Velocity and acceleration vector equations are found from time derivatives
of the vector position equation.


(d)
At time t =
sec, the position, velocity and acceleration vectors are, in
both cartesian and polar form:

(e)
The angle, b,
between the v and
a vectors, as shown above, is 9.30°. (f)
The normal and tangential components of acceleration (shown above)
are easily found once b
is known. Note:
The angle b
between v and
a
can also be found from the dot product:

Example 24: xy Plotter Parametric Equations Problem  
An
xy plotter consists of an x slider and a y slider,
with a pen located at the intersection of the two sliders.
Both sliders are capable of moving independently in time in order
to plot virtually any kind of curve.
If for a particular plot the pen position is described by x =
2t and y = 5 sin 4t , where x and y are
in feet and t is in seconds, do the following:


(a)
Eliminate time t from the parametric equations
and
to
obtain the path equation:



(c)
Velocity and acceleration components are found from time derivatives of
the parametric equations: 

At
x = 2 ft, time is t = 1 sec (since
x = 2t ). The
position, velocity and acceleration vectors are, in both cartesian and
polar form:

(d)
The angle, b,
between the
v and
a vectors, as shown above, is 8.70°. (e)
The normal and tangential components of acceleration are easily
found once b
is known. These are shown in
the picture above. 
