Environmental Engineering Reference
In-Depth Information
in general, a function of time. We differentiate with respect to time to obtain the
relative velocity,
V
ab
(t)
,
d
R
ab
(t)
d
t
V
ab
(t)
=
=
v
b
(t)
−
v
a
(t),
(1.15)
where
v
a
(t)
and
v
b
(t)
are the velocities of particles
a
and
b
.
Example 1.3.2
Consider an air-traffic controller tracking the positions of two air-
craft. The controller knows the positions and velocities of the aircraft at some instant
in time (t
0
). Assuming that the aircraft maintain their velocities, show that the
relative velocity can be used to decide whether there is a danger of a collision at
some later time.
=
Solution 1.3.2
The relative position vector at t
=
0
is
R
0
=
R
ab
(
0
)
=
r
b
(
0
)
−
r
a
(
0
).
The relative velocity is computed from the velocities of the aircraft:
V
0
=
V
ab
(
0
)
=
v
b
(
0
)
−
v
a
(
0
).
Since the aircraft have constant velocities the relative velocity is also constant and
it can be integrated with respect to time to obtain
R
ab
(t)
=
R
0
+
V
0
t.
The aircraft will collide if at some time t ,
R
ab
(t)
V
0
t. This is
a vector equation and it can only be satisfied if both the directions and magnitudes
of both sides of the equation are the same. Clearly we can only obtain a solution for
t>
0
if
R
0
and
V
0
are
anti
-
parallel
i.e. if
R
0
=
=
0
, i.e. when
R
0
=−
V
0
n
,whereR
0
and
V
0
are positive magnitudes and
n
is a unit vector. If the vectors are anti-parallel,
the collision time is R
0
/V
0
.
R
0
n
and
V
0
=−
In the previous example, we worked entirely in the frame of reference in which
the air traffic controller is at rest. It is tempting to identify the relative velocity
V
ab
also as the velocity of the aircraft
b
relative to the pilot of aircraft
a
. Strictly speak-
ing we have not proved this:
V
ab
is the velocity of
b
relative to
a
as determined
by the air traffic controller, not by the pilot of aircraft
a
. In classical mechanics,
where time is universal, the two are equivalent and specifying the relative velocity
between two bodies does not need us to further specify who is doing the observ-
ing. That the assumption of universal time enters into this matter can be seen by
exploring the expression
V
ab
(t)
d
R
ab
(t)
d
t
. Whose time is represented by
t
?Thatof
the air-traffic controller or that of the pilot in aircraft
a
? If we accept the concept
of absolute time then it doesn't matter and both record the same relative velocity.
But we really ought to recognise that the assumption of universal time is just that:
an assumption. This is not an irrelevant matter for, as we shall see in Part II, the
universality of time breaks down, becoming most apparent when relative velocities
start to approach the speed of light.
=
