Example 4-2: A cannon of mass m , resting on a frictionless surface, fires a ball of
C
mass m . The ball is fired horizontally. The muzzle velocity is v . Find the velocity of
B
M
the ball and the recoil velocity of the cannon.
NOTE: This is an example of an anti-collision problem. It also involves the concept of relative
velocity. The muzzle velocity is the relative velocity between the ball and the cannon. It is the
velocity at which the two separate. If the velocity of the ball relative to the ground is v ′B to the
right, and the velocity of the cannon relative to the ground is v ′C to the left, then the velocity of
the ball relative to the cannon, also known as the muzzle velocity of the ball, is v
= v ′ +v ′ . In
M
B
C
cases not involving guns or cannons one typically uses the notation v
for "relative velocity" or,
rel
relating to the example at hand, v
for "velocity of the ball relative to the cannon."
BC
BEFORE
AFTER
v ′
v ′
C
m
B
C
m
B
p
Σ
= Σ
→ =
p′
→
→
0 = −m v ′ + m v ′
(1)
C C
B B
Also, from the definition of muzzle
velocity:
v =v ′ +v ′
M
B
C
v ′ =v −v ′
(2)
C
M
B
Substituting this result into equation (1)
yields:
0 = −m (v
−v ′ ) + m v ′
C
M
B
B B
0 = −m v
+ m v ′ + m v ′
C M
C B
B B
m v ′ + m v ′ = m v
mC
′
C
B
B B
C M
v
=v −
v
C
M
M
m + m
C
B
(m + m )v ′ = m v
C
B
B
C M
(m + m )v
− m v
C
B
M
C M
v ′ =
m
C
+
C
v ′ =
v
m
m
C
B
B
M
m + m
C
B
m v
+ m v − m v
C M
B M
C M
v ′ =
C
m + m
C
B
Now substitute this result into equation (2) above. This yields:
mB
v ′ =
v
C
M
m + m
C
B
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