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To find the velocity vector as a function of time, we need to
integrate the acceleration vector with respect to time,
v=
adt
=
b–ct
2
m
x
+–b+ct
2
m
y
dt
=
1
2
m
b–ct
x
+–b+ct
y
dt
A vector function can be integrated component by component, so
this can be broken down into two integrals,
v=
x
2
m
b–ct
dt
+y
2
m
–b+ct
dt
=
bt–
1
2
ct
2
2
m
+const.#1
x
+
–bt+
1
2
ct
2
2
m
+const.#2
y
Here the physical significance of the two constants of integration
is that they give the initial velocity. Constant #1 is therefore zero,
and constant #2 must equal v
o
. The final result is
v=
bt–
1
2
ct
2
2
m
x
+–bt+
1
2
ct
2
2
m
+v
o
y
.
Summary
The velocity vector points in the direction of the object’s motion. Relative motion can be described by
vector addition of velocities.
The acceleration vector need not point in the same direction as the object’s motion. We use the word
“acceleration” to describe any change in an object’s velocity vector, which can be either a change in its
magnitude or a change in its direction.
An important application of the vector addition of forces is the use of Newton’s first law to analyze me-
chanical systems.
Summary