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1
Unit 3
Momentum
3.1.
Conservation of Momentum
3.2
Collisions
3.3
Impulse
3.4
Coefficient of restitution (
e
)
3.1. Conservation of Momentum
Consider that we are performing a collision experiment with two particles (not necessary
identical particles) on a twodimensional plane, say, smooth table. If the initial velocity
vectors of the two particles were labeled as
1
u
G
and
2
u
G
respectively, then after collision, their
velocity were found to be
1
v
K
and
2
v
K
respectively.
•
The theory behind the collision
During the collision, the forces act on each other are with the same magnitude but opposite in
direction. This is the Newton’s third law, it is about the action and reaction forces. They are
always opposite in directions but they have the same magnitudes (e.g.
12
FF
=−
G
G
). Hence, we
have
t
v
m
t
v
m
∆
∆
−
=
∆
∆
2
2
1
1
G
G
2
2
1
1
v
m
v
m
G
G
∆
−
=
∆
m
2
m
2
m
1
m
1
1
u
G
2
u
G
1
v
K
2
v
K
1
1
1
u
v
v
G
K
K
−
=
∆
2
2
2
u
v
v
G
K
K
−
=
∆
Before Collision
After Collision
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Substituting
∆
v
1
,
∆
v
2
and rearrange the equation, we obtain
)
(
)
(
2
2
2
1
1
1
u
v
m
u
v
m
G
G
G
G
−
−
=
−
or
11
1
2 2
2
()
(
)
0
mv u
−
+−
=
G
GG
G
.
That is,
2
2
1
1
2
2
1
1
v
m
v
m
u
m
u
m
G
G
G
G
+
=
+
The above expression is the conservation of momentum. Define the momentum of the
particle as
p
G
, where
p
mv
=
. We can rewrite the above equation as
constant
=
∑
i
i
p
K
or in
another form
0
=
∆
∑
i
i
p
K
.
•
Experimental facts
After performing numerous trials with different initial velocities and final velocity being
measured, it was found that:
(1)
1
v
K
∆
is always in opposite direction of
∆
K
v
2
(2)
1
2
v
v
∆
=
∆
K
K
constant
We can repeat the experiment by changing different particles and we found that different
particles have different degree of resistance to change its magnitude of the velocity after the
collision.
We can check that the constant is given by the ratio of
m
2
and
m
1
:
1
2
2
1
m
m
v
v
=
∆
∆
K
K
where
m
1
and
m
2
are then called the inertia mass of the particles, which is a measure of the
resistance to change the velocity magnitude during an interaction with another particle.
From
this experiment, we also discover a conservation law if we define a physical quantity called
‘momentum’ by:
v
m
p
K
K
=
.
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 Spring '08
 Prof

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