78
2 Particle Dynamics
Fig. 2.28
Using (1) and (2) and simplifying
v 1 =
v
m 1 + m 2
m 2
1 + m 2
2 + 2m 1 m 2 cos θ
(4)
2.28 (a) Let the initial momentum of one object be p. After scattering let the
momenta be p 1 and p 2 , with the angle θ between them, Fig. 2.29.
p = p 1 + p 2 (momentum conservation)
∴ ( p · p) = p
2
= ( p 1 + p 2 ) · ( p 1 + p 2 )
= p 1 · p 1 + p 2 · p 2 + p 1 · p 2 + p 2 · p 1 = p
2
1 + p
2
2 + 2 p 1 + p 2
(1)
p 2
2m
=
p 2
1
2m
+
p 2
2
2m
(energy conservation) or p
2
= p
2
1 + p
2
2
(2)
Combining (1) and (2), 2 p 1 · p 2 = 0
∴ p 1 and p 2 are orthogonal
Fig. 2.29a
Fig. 2.29b