Environmental Engineering Reference
In-Depth Information
m
1
m
1
m
2
v
1
v
2
m
2
(a)
(b)
v
′
1
v
′
2
m
1
m
2
(c)
Figure 3.8 The collision between two particles in the lab frame is considered in three stages.
At times long before (a) and long after (c) the collision the particles are far enough apart that
we can ignore their mutual interaction and both may be treated as isolated particles. During
the interaction stage (b) the particles interact and their velocity vectors are continuously
changing.
v
1
c
=
v
1
−
V
v
2
c
=
v
2
−
V
,
(3.32)
where
v
1
c
and
v
2
c
are the velocities in the new frame of reference. We assert that
the total momentum in this frame is zero, i.e.
m
1
v
1
c
+
m
2
v
2
c
=
0
,
(3.33)
and solve for
V
to obtain
m
1
v
1
+
m
2
v
2
V
=
.
(3.34)
+
m
1
m
2
Notice that the form of this equation is reminiscent of the expression for the
centre-of-mass vector, Eq. (2.27). In fact, for two colliding particles with position
vectors
r
1
and
r
2
the time-derivative of the position of the centre-of-mass is
m
1
d
r
1
d
R
d
t
=
1
m
1
+
m
2
d
r
2
d
t
d
t
+
=
V
.
(3.35)
m
2
Thus we see that zero-momentum frames are characterized by the fact that they
are those frames in which the centre of mass is at rest. We have used the plural
