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Gaining Momentum:
Chapter 9
New Vocabulary: Linear Momentum


P  mV
Impulse (force times time)
Major Concepts of Chapter 9:
•Linear momentum is conserved. It remains the same
in the absence of an external force.
•Linear momentum can be changed by an impulse (a
force acting for a period of time).
•Linear momentum is unchanged in collisions
•There are two types of collisions: inelastic and
elastic. Linear momentum is conserved in BOTH.
•An “elastic” collision is one in which the objects
“bounce”, and energy is conserved.
•An “inelastic” collision is one in which the objects
stick together, and energy is lost to heat.
Why do we need momentum?
• Momentum is useful because it remains
unchanged in the absence of external forces
• The total momentum of a system of moving
objects is the sum of the individual momenta



PTOT  m1V1  m2V2  ...

  miVi
i
Note that momentum is a VECTOR. It has magnitude AND direction.
Momentum changes with an external force.

P 
F
t
Note: is for the
case where mass
is constant.
This is a more accurate statement of Newton’s second law.
An IMPULSE can cause a change in momentum.
 

P  Ft  I
Check the units:
Momentum is mass X velocity = kg – m/s
Impulse is mass X acceleration X time = kg-m/s^2 *s
Newton’s Second Law


Ptot
Fext 
t

mv 

t

v
m
t

 ma
Most general form.
If mass is constant….
Constant mass form.
Changes in momentum.
Case A (inelastic collision)

Pi  mvyˆ

Pf  0
 
P  Pf  Pi   mvyˆ
Case B (elastic collision)

Pi  mvyˆ

Pf  mvyˆ
 
P  Pf  Pi  2mvyˆ
-Pi
P
Pi
Pf
P
Case B
Pf
Acting on Impulse: Change of Momentum

Pi  mvi xˆ

Pf  mv f xˆ
 
P  Pf  Pi   mv f  vi xˆ
P  I  FavgT
Momentum Conservation
• In the absence of an external force, linear momentum of
a system of objects does not change.
P  I  FavgT
So, if the impulse (external force) is zero,
The total momentum of a system of objects is
the sum of the individual momenta.
 
P  Pf  Pi  0


Pf  Pi



PTOT  m1V1  m2V2  ...

  miVi
i
Elastic collision: make a prediction.
We will do this experiment in a moment. Assume m1 = m2.
If the initial speed is V, what are the final speeds, V1, V2?
Elastic Collision: M1 = M2. The initial velocity of M1 is +V;
M2 is zero. The final velocities of M1 and M2, respectively,
are:
1.
2.
3.
4.
Zero, V/2
-V, V
Zero, V
V/2, V/2
Inelastic Collision: Make a prediction
Inelastic Collision: M1 = M2. The initial
velocity is V. The final velocity is:
1.
2.
3.
4.
2V
Zero
V
V/2
A matter of the point of view….
• Imagine how a collision looks, when viewed from
a video camera that is moving.
“Real world.”
2v
M
M
V
Here’s what the film sees….
“Film world.”
v
v
M
M
Before collision, the film shows two blocks of
same mass approaching at the same speed.
“Real world.”
M
2v
M
V
In the film, two identical masses approach velocity
+V and –V respectively. After the collision, the
velocity is
1.
2.
3.
4.
Zero, Zero
-V, +V
-V/2, +V/2
+V, -V (no change)
Here’s what the film sees….
“Film world.”
v
v
M
M
Before collision, the film shows two blocks of
same mass approaching at the same speed.
After the collision, the blocks move apart with equal speed.
v
v
M
M
If that’s what the film “saw”, what happened in the “real world”?….
“Film world.”
After the collision, the blocks move apart with equal speed.
v
v
M
M
V
VRe al  VFilm  VCamera
“Real world.”
2v
V=0
M
M
Actually, both situations describe “real” physics. Momentum is conserved in both cases.