Barrons AP Calculus - David Bock

This is the net change in position from t = 0 to t = 4, sometimes referred to as “position shift.” Here

it indicates the particle ended up 32 units to the right of its starting point.

EXAMPLE 3

The acceleration of an object moving on a line is given at time t by a = sin t; when t = 0 the object

is at rest. Find the distance s it travels from t = 0 to

SOLUTION: Since it follows that

Also, v(0) = 0 yields C = 1. Thus v(t) = 1 − cos t; and since cos t 1 for all t we see that v(t) 0

for all t. Thus, the distance traveled is

B. MOTION ALONG A PLANE CURVE

BC ONLY

In Chapter 4, §K, it was pointed out that, if the motion of a particle P along a curve is given
parametrically by the equations x = x(t) and y = y(t), then at time t the position vector R, the velocity
vector v, and the acceleration vector a are:

The components in the horizontal and vertical directions of R, v, and a are given respectively by the
coefficients of i and j in the corresponding vector. The slope of v is its magnitude,

is the speed of the particle, and the velocity vector is tangent to the path. The slope of a is The
distance the particle travels from time t 1 to t 2 , is given by