82
2 Particle Dynamics
Cancelling m and reorganizing (1) and (2)
v
cos β = v 0 − v cos θ
(3)
v
sin β = v sin θ
(4)
Dividing (4) by (3)
tan β =
v sin θ
v o − v cos θ
(5)
If the collision is elastic, kinetic energy must be conserved:
1
2
mv
2
0 =
1
2
mv
2
+
1
2
mv
2
(6)
or
v
2
0 = v
2
+ v
2
(7)
Squaring (3) and (4) and adding
v
2
= v
2
+ v
2
0 − 2v 0 v cos θ
(8)
Eliminating v 2 between (7) and (8) and simplifying
v = v 0 cos θ
2.34 The momenta of β and 14 N are indicated in both magnitude and direction
in Fig. 2.31. The resultant R of these momenta is given from the diagonal
AC of the rectangle (parallelogram law). The momentum of u is obtained by
protruding CA to E such that AE = AC:
Fig. 2.31 Decay of 14 C at
rest