The purpose of this lab is to give you practice with parametrizing curves in the plane and in visualizing parametric curves as representing motion of particles.

A parametric curve in the plane can be defined as an ordered
pair, , of functions, with representing the **x**
coordinate and the **y** coordinate. Parametric curves arise
naturally as the solutions of differential equations and often
represent the motion of a particle or other mechanical systems. They
also often arise in studying oscillations in electrical circuits. For
this lab, however, we want to think of parametric curves as
representing motion of a particle in space.

For example, neglecting air resistance, the position of a projectile fired from the origin at an initial speed of and angle of inclination is given by the parametric equations

where **t** is time and **g** is the acceleration due to gravity.

Graphically, a parametric curve can be represented several ways. One simple
way is to plot the component functions, and ,
individually versus the independent variable **t**. Another way is to
plot the set of points .
This gives you the curve along which the particle moves, but
information on how it moves has been lost. On the other hand, plotting
the component functions individually makes it hard to see how the
particle is actually moving.

To help you to visualize parametric curves as representing motion, a
Maple routine called `ParamPlot` has been written. It uses the
Maple `animate` command to actually show the particle moving along
its trajectory. To use the `ParamPlot` command, you will first have
to load the `CalcP` package with the command.

> with(CalcP):

The syntax for `ParamPlot` is very simple, as shown by the
following examples. Try them! They use the Maple `animate`
command, which has controls similar to those on a VCR, allowing you to
slow down or speed up the playback, or even to put it into a
continuous loop mode. The last example shows how to animate more than
one curve at a time.

> ParamPlot([t,t^2],t=-2..2);

> ParamPlot([-t, t^2],t=0..2*Pi);

> ParamPlot([t,t^2],t=-1..2);

> ParamPlot([cos(t), sin(t)],t=0..2*Pi);

> ParamPlot([cos(t), sin(2*t)],t=0..2*Pi);

> ParamPlot([t^2, t^3-t], t=-2..2);

> ParamPlot([t,t^2],[-t,2*t],t=-2..2);

Given a curve defined by the graph of a function , there are
an infinite
number of ways of representing this curve parametrically,
corresponding to different motions on the curve. About the simplest
way of parametrizing is with the pair , which
traverses the curve from left to right as **t** increases. One can
reverse the direction of motion by changing to the parametrization
.

By restricting **t** to an interval , you can get a parametric
description of a portion of the curve. For example, the right half of
the parabola would result from for .

More complicated parametrizations of can be obtained with
parametric curves of the form . By choosing
appropriately, for example, you can make the particle stop and turn back on the
curve. For example, suppose that the curve to be parametrized is the
graph of the function **y=2x**. The following examples give three
different parametrizations of this curve.

> ParamPlot([sin(t),2*sin(t)], t=0..4*Pi);

> ParamPlot([t-1,2*(t-1)],t=0..2);

> ParamPlot([(1-t)^2,2*(1-t)^2],t=0..3);

- Suppose that a cannon can fire a projectile at an initial
velocity of 400 feet per second. Neglecting air resistance, what angle
of elevation should be used to hit a target at the same elevation 500
yards away? Explain how your result was obtained.
- Lissajous figures. Investigate the family of parametric curves
where

**n**and**m**are positive integers. Explain what you see. ( Due to limits on`ParamPlot`, you will probably have to keep**n**and**m**smaller than about 5.) - The last Maple example, concerning the curve
**y=2x**, provided three different parametrizations. Match the parametrizations with the descriptions below. Justify your choices.- Starts at the point . Moves to the left initially, but reverses direction once
- Starts at the origin, then oscillates between the points and .
- Starts at the point and then moves to the right along the line.

- Given a function , find conditions on that are
necessary to make the parametric description do the
following. Give conditions on and its derivatives, as well as
examples, and make sure that you explain your reasoning.
- Traverse the curve from left to right as
**t**increases. - Reverse direction once.

- Traverse the curve from left to right as
- Suppose that at time zero, flight 12 is at the point
at an altitude of 30,000
feet and traveling
Northeast at 420 mph and that flight 33 is at the same altitude, but
is traveling due west at a speed of 388 mph. At time zero, flight 33
is at the point .
- What is the minimum distance between flights 12 and 33, and at what time does it occur?
- Suppose that FAA regulations require that the closest approach of any two aircraft must be greater than 2000 feet. Can you change the speed of flight 33 so that this regulation is satisfied? What is your new speed, and what is the closest approach distance?

Mon Jun 26 15:29:53 EDT 1995