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.