88
2 Particle Dynamics
Use (3) and (4) in (1) to get
1 + cot
2
θ = cos ec
2
θ =
M 2
m 2
∴ sin θ max =
m
M
or θ max = sin
−1
m
M
2.44 Momentum of the bullet before collision = momentum of the block + bullet
system immediately after collision, Fig. 2.34:
Fig. 2.34
mv = (m + M)V
(1)
where V is the initial speed of the block + bullet system. The kinetic energy
of the system immediately after the impact is
K =
1
2
(m + M)V
2
(2)
Due to the impact, the pendulum would swing to the right and would be raised
through the maximum height h vertically above the rest position of the pendulum, Fig. 2.34. At this point, the kinetic energy of the pendulum is entirely
converted into gravitational potential energy:
1
2
(m + M)V
2
= (m + M)gh
(3)
∴ V =
2gh
(4)
Using (4) in (1)
v =
1 +
M
m
2gh
(5)