Barrons AP Calculus - David Bock

and its magnitude is the vector’s length:

where we have used ax and ay for respectively.

Examples 28 and 29 are BC ONLY.

EXAMPLE 28

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

(a) Find a single equation in x and y for the path of the particle and sketch the curve.

(b) Find the velocity and acceleration vectors at any time t, and show that a = −R at all times.

(c) Find R, v, and a when (1) (2) t 2 = π, and draw them on the sketch.

(d) Find the speed of the particle and the magnitude of its acceleration at each instant in (c).

(e) When is the speed a maximum? A minimum?

SOLUTIONS:

(a) Since therefore

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.

(b) We have

The acceleration, then, is always directed toward the center of the ellipse.

(c) At

At t 2 = π,

The curve, and v and a at t 1 and t 2 , are sketched in Figure N4–18, below.

(d) At At t 2 = π,