Environmental Engineering Reference
In-Depth Information
This will lead to more complicated expressions for velocity and acceleration in
polar co-ordinates than are obtained for Cartesian co-ordinates, as will be seen in
the next section.
1.3 VELOCITY AND ACCELERATION
A particle is in motion when its position vector depends on time. The Ancient
Greek philosophers had problems accepting the idea of a body being both in motion,
and being 'at a point in space' at the same time. Zeno, in presenting his 'runner's
paradox', divided up the interval between the start and finish of a race to produce
an infinite sum for the total distance covered. He argued that before the runner
completes the full distance (
l
) he must get half-way, and before he gets to the end
of the second half he must get to half of that length and so on. The total distance
covered can therefore be written as the infinite series
l
1
.
1
4
+
1
8
+···
2
+
Zeno argued that it would be impossible for the runner to cover all of the
sub-stretches in a finite time, and would therefore never get to the finish line. This
contradiction forced him to decide that motion is impossible and that what we
perceive as motion must be an illusion. We now know that the resolution of this
paradox lies in an understanding of calculus. As the series continues, the steps get
shorter and shorter, as do the time intervals taken for the runner to cover each
step and we tend to a situation in which a vanishingly short distance is covered in
a vanishingly small time.
Assuming that the position is a smooth function of time, we define the velocity as
r
(t
.
d
r
(t)
d
t
+
t)
−
r
(t)
v
(t)
=
=
limit
t
(1.10)
t
→
0
+
Notice that it involves a difference in the position vector at time
t
t
and at
time
t
. This difference, divided by the time interval
t
, only becomes the velocity
in the limit
that
t
goes to zero. Thus the velocity is defined in terms of an
infinitesimally small displacement divided by an infinitesimally small time interval.
Notice that the vector nature of
v
follows directly from the vector nature of
r
(t
r
(t)
, which differs from
v
only by division by the scalar
t
.Oftenit
is useful to refer to the magnitude of the velocity; this is known as the speed
v
,i.e.
+
t)
−
=
|
|
v
v
.
With the notion that the ratio of two infinitesimally small quantities can be a
finite number, we return to the Runner's Paradox. Zeno's argument does not rely
on the particular choice of infinite series stated above. So we can simplify things
by instead using a series made of equal-length steps. First we divide
l
up into
