Environmental Engineering Reference
In-Depth Information
Q
> 0
Q
< 0
p
1
C
p
′
1
C
p
1
C
q
C
q
C
p
′
1
C
p
′
2
C
p
2
C
p
′
2
C
p
2
C
Figure 3.10
Momentum diagrams for inelastic collisions with
Q>
0and
Q<
0.
collides with the second, stationary car. Calculate: (a) the final velocities of the
cars in the lab frame assuming an elastic collision; (b) the Q-value of an inelastic
collision if the cars stick together.
Solution 3.3.1
(a) We shall perform the calculation in the lab frame. The total
momentum is mv and the law of momentum conservation implies that the momenta
after the collision must satisfy
mv
1
+
mv
2
=
mv,
i.e.
v
1
+
v
2
=
v.
(3.47)
The collision is elastic so
1
2
mv
1
1
2
mv
2
1
2
mv
2
,
+
=
i.e.
v
1
v
2
v
2
.
+
=
(3.48)
Squaring Eq. (3.47) and subtracting Eq. (3.48) gives
2
v
1
v
2
=
0
,
which means that one of the cars must have zero final velocity. Eq. (3.47) then
implies that the other car must have final velocity v. Since Nature does not permit
the cars to pass through each other, the car with final velocity v must be the one
that was originally stationary. Thus, in terms of early- and late-stage velocities we
have, e.g., v
1
=
0
; v
1
=
0
v
2
=
vv
2
=
v.
