Barrons AP Calculus

(a)

(b)

(c)

(d)

(e)

(a)

(b)

Acceleration    vector

The acceleration     vector a    is  ,   and     can     be  obtained    by  a   second

differentiation of the components of R. Thus

,

and its magnitude   is  the vector’s    length:

,

where   we  have    used    ax  and ay  for  ,  respectively.

Example 28 __

A particle moves according to the equations x = 3 cos t, y = 2 sin t.

Find    a   single  equation    in  x   and y   for the path    of  the particle    and sketch  the

curve.

Find    the velocity    and acceleration    vectors at  any time    t,  and show    that    a =

−R  at  all times.

Find    R,  v,  and a   when    (1)  ,  and draw    them    on  the sketch.

Find    the speed   of  the particle    and the magnitude   of  its acceleration    at  each

instant in  (c).

When    is  the speed   a   maximum?    A   minimum?

SOLUTIONS:

and  the     particle    moves   in  a   counterclockwise    direction   along   an  ellipse,

starting,   when    t   =   0,  at  (3, 0)  and returning   to  this    point   when    t   =   2π.

We  have

R   =   3   cos t,  2   sin t