COALITION MATH 151, FALL 1997

        Group II (S. A. Fulling, assisted by Cary Lasher)

                              Day 6.2


    1.  Projections (decomposition of vectors into components)
(THIS SHOULD HAVE BEEN DONE ON THE PREVIOUS DAY.)

        A.  Worked example with diagram and arithmetic; add a 
        second vector to get components with respect to an 
        orthonormal basis.

        B.  Team RAT:  Similar problem at end of vectors Web 
        page (3 min to e-mail answer; all team members should 
        understand) 
STUDENTS APPEARED VERY UNCOMFORTABLE WITH THIS, SO I CAME BACK TO 
IT ON THE FOLLOWING DAY WITH A LECTURE ON ROTATED COORDINATE 
SYSTEMS.

    2.  Parametric equations

        A.  No big theorems or formulas in this subject; you need 
        experience and a bit of basic terminology (e.g., tangent 
        vector). 

        B.  Activity:  (team transparencies)
    
            i.  Write a parametric representation of the circle 
            with center (2,1) and radius 4.  (Let the parameter 
            t vary from 0 to 2*Pi and have the "motion" start at 
            the rightmost point on the circle.) 

            ii.  Find the tangent vector to this parametrized 
            circle as a function of t, and at the point (2,5).
            (Sketch.)
RAN OUT OF TIME AT THIS POINT; iii DONE NEXT DAY.

            iii.  Suppose the parametrization is a real physical 
            motion (so tangent vector = velocity vector).  Find 
            the acceleration vector as a function of t, and at 
            the 4 extreme points of the circle.  Sketch, and 
            describe qualitatively. 
ALTHOUGH SOME STUDENTS KNEW THAT THE ACCELERATION POINTS TO THE 
CENTER, IT WAS NOT POSSIBLE TO FIND A TEAM THAT WOULD SO ARGUE, 
CONVINCINGLY AND IN GENERAL.

POSTPONED THIS UNTIL THE DAY ON THE CHAIN RULE:
        C.  Now suppose the motion is at a different speed, so 
        the formulas involve sin(3t), etc.  How can we calculate 
        the velocity and acceleration?  Look forward to next 
        week's topics:

            i.  Math:  Chain rule (assert for the example)

            ii.  Engineering:  Units and dimensions.  Let u be
            elapsed time in "Texas seconds" that are only 1/3 as 
            long as normal New York seconds.  Thus u = 3t, and
            in terms of u we have the problem previously solved.
            Then, converting from feet per Texas second to feet 
            per second is just like converting highway costs from 
            dollars per foot to dollars per yard:  multiply by 3.
            For acceleration, we must do this again.

    3.  Comment on test results and midterm grades