Be Careful -- you may wonder why
we just don't consider the position of the particle at 2 times and deduce subsequent motion from that.
the same reason wagon wheels turn backward in old movies.
Consider the following animation in which circular motion
viewed from different time intervals shows in one case the particle moving counterclockwise and in another it appears to be
Consideration of the instantaneous rates of change eliminates this paradox
non smooth example
Consider x(t) =cos2(t) and y(t) =cos(t).
1. By inspection we see y = x2.
2. Initially the particle is at (1,1)
3. x'(t) = -2cos(t)sin(t)
= -sin(2t) y' (t) =-sin(t)
We see the derivatives are simultaneously 0 ar integer multiple of π
We'll analyze the derivatives on the intervals 0 to π and then π to 2π.
Just after t = 0 the
derivaties <0 so the particle is moving to the left and down. At π/2 the particle starts moving up but is still moving
to the left. At t =π the particle comes to a stop and just after π starts moving down and to the right returning to
(1,1) at t=2π.
Noe at 3π/2 the particle starts moving up but the motion is still to the right.
motion then repaeats over every 2π interval.
Non Smooth Example-Animation
Here's one for you to try x(t)
=sin(t) y(t) =sin(t)
See Answer below
An Example with pts of non-differentiability
Let x(t) = |cos(t)| and y(t) = sin(t)
We still have circular motion,however at odd multiples
of π/2 x '(t) does not exist and instaneously changes signs,therefore the particle "bounces off the wall"
at these times
Consider the 2
sets of parametric eqns :
x(t) = t y(t) = t and x(t) =et - 1 and
y(t) = et - 1
1. Both travel on y = x
2. Both start at (0,0)
3. Both are smooth and represent
a particle moving up and to the right.
The difference is speed
The best way to see this is in terms
let r(t) = x(t) i + y(t) j
then v(t) = x'(t)i +y'(t)j
and since the speed ds/dt = ||v(t)|| it follows the speed is √(x'2
Therefore for our first particle the speed ds/dt = √2 and for the second is et√2
It follows the distance traveled is then ∫√(x'2 +y'2)dt integrated from t1
to t2 .
See animation below
Same Path Different Speeds
In this example we'll consider
a planet-moon system.
Suppose a planet orbits a star every 365 days at a distance of 9.3 million miles.
Then x(t) = 9.3cos(2π/365t) y(t) = 9.3sin(2π/365t)
At the same time a moon orbits the
planet every 28 days at a distance of 2.5 million miles
Then x(t) = 9.3cos(2π/365t) + 2.5cos(2π/28t) y(t)
= 9.3sin(2π/365) + 2.5sin(2π/28t)
See the animation below
In this example we consider a mosquito walking
on a rotating disk.
In the first animation the mosquito
approaches the center at a constant rate.
second animation the mosquito has become disoriented. Leaving the center
It alternates between moving toward the edge and back to the center.
Notes- Mosquito on a Rotating Disk
Animation - Mosquito on a Disk
Animation - Disoriented Mosquito on a Disk
A cycloid is the curve traced out by a fixed point
on a circle as the circle rolls along the x-axis. The following notes and animation develops and demonstraes the parametric
equations of the cycloid.
Notes - The Cycloid
Animation - The Cycloid
A hypocycloid is the curve traced out by a fixed point
on a circle that rotates inside a larger circle--you can think of taking the cycloid in the previous example and folding a segment of the x-axis into a circle
following are animations of various hypocycloids.
For an excellent discussion
see the work of Nick Whitman:
Dsicussion by Nick Whitman on hypocycloids
Animation - Hypocycloid Ratio of 3 to 1
Animation - Hypocycloid - Ratio of 5 to 1
Animation - Hypocycloid - Ratio of 11 to 2.7
The following links are to the pages on supplemental exercises for parametric equations
in 2 space and a discussion of parametric equations in 3 space respectively
Supplemental Exercises for parametric eqns in 2-space
Parametric Equations in 3 space
The Smart Bunny-A very short story by Kurt Vonnegut Jr.