Calculus AB and Calculus BC

CHAPTER 4 Applications of Differential Calculus

K. MOTION ALONG A CURVE: VELOCITY AND ACCELERATION VECTORS

If a point P moves along a curve defined parametrically by P(t) = (x(t), y(t)), where t represents time, then the vector from the origin to P is called the position vector, with x as its horizontal component and y as its vertical component. The set of position vectors for all values of t in the domain common to x(t) and y(t) is called vector function.

A vector may be symbolized either by a boldface letter (R) or an italic letter with an arrow written over it Image The position vector, then, may be written as Image or as Image In print the boldface notation is clearer, and will be used in this book; when writing by hand, the arrow notation is simpler.

The velocity vector is the derivative of the vector function (the position vector):

Image

Alternative notations for Image are vx and vy, respectively; these are the components of v in the horizontal and vertical directions, respectively. The slope of v is

Image

which is the slope of the curve; the magnitude of v is the vector’s length:

Image

BC ONLY

Thus, if the vector v is drawn initiating at P, it will be tangent to the curve at P and its magnitude will be the speed of the particle at P.

The acceleration vector a is Image and can be obtained by a second differentiation of the components of R. Thus

Image

and its magnitude is the vector’s length:

Image

where we have used ax and ay for Image 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) Image (2) t2 = 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 Image therefore

Image

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

Image

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

(c) At Image

Image

At t2 = π,

Image

The curve, and v and a at t1 and t2, are sketched in Figure N4–18, below.

(d) At Image

At t2 = π,

Image

Image

FIGURE N4–18

(e) For the speed |v| at any time t

Image

We see immediately that the speed is a maximum when Image and a minimum when t = 0 or π. The particle goes fastest at the ends of the minor axis and most slowly at the ends of the major axis. Generally one can determine maximum or minimum speed by finding Image setting it equal to zero, and applying the usual tests to sort out values of t that yield maximum or minimum speeds.

EXAMPLE 29

A particle moves along the parabola y = x2x with constant speed Image Find v at (2,2).

SOLUTION: Since

Image

and

Image

Relation (3) holds at all times; specifically, at (2, 2), Image so that vx = ±1. From (1), then, we see that vy = ±3. Therefore v at (2, 2) is either Image The former corresponds to counterclockwise motion along the parabola, as shown in Figure N4–19a; the latter to clockwise motion, indicated in Figure N4–19b.

Image

FIGURE N4–19a

Image

FIGURE N4–19b