For the example with , answer the following questions.
For which value(s) of t is the speed a maximum? What is the
maximum speed?
For which value(s) of t is the speed a minimum? What is the
minimum speed?
For which value(s) of t is the curvature a maximum? What is
the maximum curvature?
For which value(s) of t is the curvature a minumum? What is
the minimum curvature?
Consider the curve .
Show that the image curve of is an ellipse. That
is, find values of a and b such that the component functions of
satisfy the equation
Find the values of t, , for which the
curvature of is a maximum or a minimum and identify which
is which. What is the maximum and minimum curvature?
Find the values of t, , for which the
speed for the trajectory is a maximum or a minimum and
identify which is which. What is the maximum and minimum speed?
Consider . What is the
normal vector at ? At ? What about at t=0? What
goes wrong?
Consider the helix
where A, b, and are parameters.
Plot for a few values of the parameters. Try to
identify how each parameter affects the curve. A good way to do this
is with the subs command, like the following example.
> VPlot(subs({A=1,b=1,omega=1},r(t)),t=0..2*Pi);
Show, using the Speed command that the speed is
constant and describe how it depends on the parameters.
Compute the curvature and describe how it depends on the
parameters.
A hairpin turn on a roadway can be approximated as half of an
ellipse, including the major (longer) axis. If the major axis is 100
feet and the minor axis is 80 feet, what is the maximum (constant)
speed a car can go through the turn while keeping the centripetal
acceleration less than ?