(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