This

**preview**has**blurred**sections. Sign up to view the full version! View Full DocumentImpulse and Collisions –
Ch 9
Male rams butt heads at high speeds in a ritual to assert their
dominance. How can the force of this collision be minimized so as
to avoid damage to their brains?

Linear Momentum
Momentum is a vector; there is a direction and magnitude.
Momentum
= mass x
velocity
p
= m
v
In general, we must worry about the x, y, and z components of
momentum
If the particle is moving with in an arbitrary
direction then p must have three components which are
equivalent to the component equations:
p
x
= mv
x
p
y
= mv
y
p
z
= mv
z
Newton called m
v
the
quality of motion
.

v
v
v
A block and a ball of the same
mass and downward speed v hit
the floor.
The block falls and “sticks”
to the floor.
x
y
The ball rebounds with
the same velocity it hit
the floor with.
Change in Momentum

v
x
y
Change in Momentum
We only have to
consider the
momentum in the y
direction.
Block
∆
p
y
= p
y,f
– p
y,i
= 0 – m(-v) = mv
The change in momentum is mv upward.
Ball
∆
p
y
= p
y,f
– p
y,I
= mv – m(-v) = 2mv
The change in momentum is 2mv upward.
Be careful about the vector nature of momentum!!!

Linear Momentum
Newton's Second Law, which we have written as
F
= m
a
can also be written in terms of momentum,
F
= m
a
= m (d
v
/dt)
= d(m
v
)/dt
F
= d
p
/dt
The force acting on an object equals the time rate of change of
the momentum of that object.

Conservation of Momentum
Consider two objects that "interact" or collide or have some effect
on each other (billiard balls).
From Newton's Third Law of motion, we know
F
12
= -
F
21
Where
F
12
is the force exerted by particle 1 on particle 2 and
F
21
is
the force exerted by particle 2 on particle 1.
In terms of momentum, this means
d
p
1
/dt = - d
p
2
/dt
d
p
1
/dt + d
p
2
/dt = 0
d (
p
1
+
p
2
) /dt = 0
d
p
Tot
/dt = 0;
p
Tot
=
p
1
+
p
2

Conservation of Momentum
Lets look at that again!
d
p
Tot
/dt = 0
where
p
Tot
=
Σ
p
=
p
1
+
p
2
= const
This statement may also be written as
p
1i
+
p
2i
=
p
1f
+
p
2f
Since the time derivative of the momentum is zero, the
total
momentum
of the system remains
constant
.
This is
Conservation of Momentum
Conservation of Momentum is essentially a restatement of
Newton's Third Law of Motion.

Conservation of Momentum
We have said that if the particle is moving with in an arbitrary
direction then
p
must have three components which are equivalent
to the component equations.
This allows us to rewrite the
conservation of momentum as
∑
∑
∑
∑
∑
∑
=
=
=
system
zf
system
zi
system
yf
system
yi
system
xf
system
xi
p
p
p
p
p
p
Conservation of momentum: Whenever two or more particles in
an isolated system interact, the total momentum of the system
remains constant.

1-D inelastic collisions
Perfectly inelastic collisions
are those in which the two objects stick
together.
Consider a mass m

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