Environmental Engineering Reference
In-Depth Information
1.3.3 Uniform acceleration
In many physical situations the acceleration does not change with time. Integra-
tion of Eq. (1.13) then gives
v
=
a
d
t
=
v
0
+
a
t,
(1.16)
where
v
0
is the velocity at time
t
=
0. Since
v
0
is a constant vector, integration
again yields
1
2
a
t
2
.
r
=
r
0
+
+
v
0
t
(1.17)
In general the vectors
r
0
,
v
0
and
a
will have different directions and each of the
vector equations, Eq. (1.16) and Eq. (1.17), is shorthand for three different scalar
equations, one for each of the three spatial components. An important simplifica-
tion occurs in situations where the velocity, acceleration and displacement are all
collinear (i.e. all in the same direction). Then we need only consider the components
of the vectors along the direction of motion, i.e.
v
=
v
0
+
at
(1.18)
and
1
2
at
2
.
r
=
r
0
+
v
0
t
+
(1.19)
Squaring Eq. (1.18) and substituting using Eq. (1.19) yields a third equation that
is often useful in solving problems that don't deal explicitly with time:
v
2
v
0
+
=
2
a(r
−
r
0
).
(1.20)
Even if
r
,
v
and
a
are not collinear then Eq. (1.18), Eq. (1.19) and Eq. (1.20) can
still be applied to each of the Cartesian components of the vectors since the basis
vectors
i
,
j
and
k
are independent of time.
As an example, let us consider the problem of projectile motion in a uniform
gravitational field. Close to the Earth's surface any object accelerates towards the
centre of the Earth. This acceleration has magnitude
9
.
81 ms
−
2
g
≈
although the exact value depends on where you are on the surface of the Earth.
The fact that all objects fall at the same rate is rather amazing, but we will defer a
discussion of that until the next chapter. Here we only want to use the result that
the acceleration is uniform, which is true so long as we stick to low altitudes and
ignore the effects of air resistance.
Example 1.3.3
Determine the path of a projectile fired with speed u at an angle θ
to the horizontal. Neglect air resistance. Use the path to determine the range of the
projectile.
