165

B. The figure shows a roller coaster car rolling down and then up under the

influence of gravity. Sketch the car’s velocity vectors and acceleration vectors.

Pick an interesting point in the motion and sketch a set of force vectors acting

on the car whose vector sum could have resulted in the right acceleration

vector.

8.4

.

Calculus With Vectors

The definitions of the velocity and acceleration components given in

chapter 6 can be translated into calculus notation as

v=dx

dt

x

+dy

dt

y

+dz

dt

z

and

a=dv

x

dt

x

+dv

y

dt

y

+dv

z

dt

z

.

To make the notation less cumbersome, we generalize the concept of the

derivative to include derivatives of vectors, so that we can abbreviate the

above equations as

v=dr

dt

and

a=dv

dt

.

In words, to take the derivative of a vector, you take the derivatives of its

components and make a new vector out of those. This definition means

that the derivative of a vector function has the familiar properties

dcf

dt

=cdf

dt

[c is a constant]

and

df+g

dt

=df

dt

+dg

dt

.[c is a constant]

The integral of a vector is likewise defined as integrating component by

component.

Section 8.4

.

Calculus With Vectors

Discussion question C.