Download MATH 1411 – Final Project

MATH 1511 – Final Project
Instructions: You must type all answers to the following questions into a Maple worksheet (which should also
contain your name). You must send me that Maple worksheet as an attachment to an email message to me
(wachsmut@shu.edu) on or before May 6, 2005. You are required to complete this assignment on your own, but you
may use Maple, and/or any type of calculator, and/or any Calc book you like.
It is perfectly okay to use Maple to help you with any calculation, but if you rather do the calculations manually, that
would be fine as well as long as you type up sufficient explanations in your Maple worksheet.
To answer the various “conjectures”, type as much as you feel is necessary to convey your ideas. Support your
conjectures with Maple plots or computations, if appropriate.
I know it is exceedingly easy to “copy” assignments like this – any plagiarism or cheating I detect it will result in 0
points for every project involved and any further action that seems appropriate. Note that all projects differ slightly.
If you have any questions, please send me electronic mail as soon as possible.
Question 1: Below are four parametric curves, and four named parametric equations. Which
curve belongs to which equation?

Lissajous curve:
x(t )  4 cos(t )
y (t )  2 sin( 2t )

Evolute of ellipse:
x(t )  cos 3 (t )
y(t )  2 sin 3 (t )

Involute of circle:
x(t )  cos(t )  t sin( t )
y (t )  sin( t )  t cos(t )

Serpentine curve:
x(t )  cot(t )
y (t )  4 sin( t ) cos(t )
Conjecture: Describe how the Lissajous curves x(t )  4 cos(t ) , y (t )  2 sin( nt ) will look for
different values of n, where n is a positive integer.
Question 2: In Greek, the word cycloid means “wheel”, the word hypocycloid means “under the
wheel”, and the word epicycloid means “upon the wheel”. You can obtain an epicycloids, for
example, by tracing a point on a small circle as it rolls around the outside circumference of a
larger circle.
A Hypocycloid H(A, B) has the following parametric equation:
A B
t)
B
A B
y (t )  ( A  B) sin( t )  Bsin(
t)
B
An Epicycloid E(A, B) has the following parametric equation:
x(t )  ( A  B) cos(t )  B cos(
A B
t)
B
A B
y (t )  ( A  B) sin( t )  Bsin(
t)
B
x(t )  ( A  B) cos(t )  B cos(
Use Maple to create the following four graphs:

H(10,3) (i.e. a hypocycloid where A = 10 and B = 3)

H(29, 9) (i.e. a hypocycloid where A = 29 and B = 9)

E(10, 3) (i.e. an epicycloids where A = 10 and B = 3)

E(29, 9) (i.e. an epicycloids where A = 29 and B = 9)
Note that the parameter t must be in a reasonably large interval, like [0,100] , to give good
pictures.
Conjecture A: Consider a hypocycloid H(A, B) and set B = 3. Describe what the impact of the
parameter A is, as A is 3, 5, 7, 11, 13, etc.
Conjecture B: Consider a hypocycloid H(A, B) and set A = 13. Describe what the impact of the
parameter B is, as B is 1, 2, 3, …, 13. What will happen if B becomes larger than A?
Question 3: The following functions are given in polar coordinates where r denotes the radius
and t denotes the angle.


Limacons: r  a  b cos(t ) or r  a  b sin( t )
Rose curves: r  a cos(nt ) or r  a sin( nt )
Use Maple to draw three Limacons using the cos function, where



a = 2 and b = 3
a = 3 and b = 3
a = 3 and b = 2
Use Maple to draw three Rose curves using the cos function, where



a = 2 and n = 3
a = 2 and n = 5
a = 2 and n = 4
Conjecture A: Describe what happens to the above pictures if I used the respective sin functions
instead of the cos functions?
Conjecture B: For a Limacon, investigate how the shape will change depending on a and b.
Conjecture C: For a Rose curve, what impact does the parameter n have on the shape of the
rose? How about the impact of the parameter a?
Extra Credit: Suppose an object of mass m is released from a balloon at very high altitude,
falling towards earth. Assume that the force of resistance Fr due to the air is directly proportional
to the speed of the object, i.e. Fr  kv . The force due to gravity is clearly Fg  mg . Both forces
work in opposite direction so that the total force F  Fg  Fr . Finally, according to Newton’s
dv
, where acceleration is the derivative of the velocity. Putting
dt
everything together we get the following differential equation for the velocity of our object:
Law we know that F  ma  m
m
dv
 mg  kv
dt
or , after dividing by m
dv
k
g v
dt
m
which is a separable differential equation.
Suppose an object of mass 1 is used, and the constant of proportionality k  2 .


Solve the above separable differential equation in this particular situation (i.e. m = 1 and
k = 2) where you may assume that the gravity constant g = 10 (approximately). Note that
you can use the condition that the object at time 0 has zero velocity (i.e. v(0) = 0) to find
an actual value for any constant of integration.
What is the limiting velocity, i.e. what is the velocity if the object falls “forever”, i.e. as a
limit where t approaches infinity
Conjecture: Is the limiting velocity of an object of mass 2 faster or slower than in the above
example where m = 1 was used. Prove your conjecture by repeating the above calculation for m
= 2.
Have a nice summer !