Barrons AP Calculus

(a)

(b)

(a)

(b)

Example 7 __

A particle P(x, y) moves along a curve so that its acceleration is given by

when t = 0, the particle is at (1, 0) with

Find    the position    vector  R   at  any time    t.

Find    a   Cartesian   equation    for the path    of  the particle,   and identify    the conic

on  which   P   moves.

SOLUTIONS:

v   =    −2 sin 2t + c 1 ,  2   cos t + c 2 ,   and since   v   =    0,2    when    t   =   0,  it  follows

that    c 1     =   c 2     =   0.  So  v   =    −2 sin 2t, 2   cos t.  Also    R   =   cos 2t + c 3 ,  2   sin t +

c 4 ;   and since   R   =   1,0 when    t   =   0,  we  see that    c 3     =   c 4     =   0.  Finally,    then,

R   =   cos 2t, 2   sin t.

BC  ONLY

From    (a) the parametric  equations   of  motion  are

x   =   cos 2t, y   =   2   sin t.

By  a   trigonometric   identity,

P   travels in  a   counterclockwise    direction   along   part    of  a   parabola    that    has its

vertex  at  (1, 0)  and opens   to  the left.   The path    of  the particle    is  sketched    in

Figure  N8−1;   note    that    −1  ≤   x   ≤   1,  −2  ≤   y   ≤   2.