Dr. Larry Dennis,

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Physics 2053C Fall 2001

• Chapter 7
• Linear Momentum
• Conservation of Momentum

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Conservation of Momentum

• Recall
• Generalized Work-Energy Theorem
• K1 U1 Wnc K2 U2
• Conservation of Energy means the total energy
  doesnt change.
• Conservation of Momentum means the total momentum
  doesnt change.

Whats Momentum????
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Momentum

• p mv

1. Momentum mass velocity
2. Momentum is a vector.
3. It is parallel to the velocity.

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Equivalent Formulation of Newtons Second Law
The rate of change of momentum of a body is
equal to the net force applied to it.
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Equivalent Formulation of Newtons Second Law
The rate of change of momentum of a body is equal
to the net force applied to it.
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Newtons Second Law Momentum

• When there is a net force Momentum changes.
• ?p ? F?t Impulse
• When there is no net force Momentum remains
  constant.
• ?p ? F?t 0

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CAPA 1 2

• A golf ball of mass 0.05 kg is hit off the tee at
  a speed of 45 m/s. The golf club was in contact
  with the ball for 5.0x10-3 s.
• Find the impulse imparted to the ball.

Impulse ?F?t ?p mvf mvo Impulse m(vf
vo) 0.05kg (45 0)m/s Impulse 2.25
kg-m/s 2.25 N-s
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CAPA 1 2

• A golf ball of mass 0.05 kg is hit off the tee at
  a speed of 45 m/s. The golf club was in contact
  with the ball for 5.0x10-3 s.
• Find the average force imparted to the golf ball
  by the club.

Impulse ?F?t ( 2.25 N-s ) Impulse/?t
?F ?F 2.25 N-s/5.0x10-3 s 450 N
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Application to Collisions
When ?p 0 then Momentum Before Collision
Momentum After Collision Mathematically this
means M1V1b M2V2b M1V1a M2V2a
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Types of Collisions
An Elastic Collision Kinetic Energy does not
change. An Inelastic Collision Kinetic Energy
changes
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Inelastic Collision CAPA 4
Momentum Before Collision Momentum After
Collision MtVtb MsVsb MtVta MsVsa
Vtb
Vta Vsa 0
Vsb
MtVtb MsVsb (MtMs) 0 MtVtb - MsVsb ? Vsb
-VtbMt/Ms
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Inelastic Collision
Momentum Before Collision Momentum After
Collision M1V1b M2V2b M1V1a M2V2a
V1b Vb
V2b 0
V1a V2a Va
M1Vb M20 (M1M2)Va
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Inelastic Collisions (cont)
M1Vb M20 (M1M2)Va M1Vb
(M1M2)Va M1/(M1M2) Vb Va
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Center of Mass Velocity

• M1V1b M2V2b ? (M1 M2 )Vcm

U V - Vcm
and V U Vcm
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Elastic Collision in the CM
U2a
U1a
M1U1a M2U2a 0
M1U1b M2U2b 0
½M1U21b ½M2U22b ½M1U21a ½M2U22a
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Elastic Collision in the CM
M1U1b M2U2b 0
M1U1a M2U2a 0
½M1U21b ½M2U22b ½M1U21a ½M2U22a
Solution Mirror Image U1a -U1b U2a
-U2b
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Elastic Collision in the CM CAPA 6-9
4.23 m/s
3.88 m/s

• A pair of bumper cars in an amusement park ride
  collide elastically as one approaches the other
  directly from the rear. One has mass m1 467 kg
  and the other mass m2 567 kg. If the lighter
  one approaches at v1 4.23 m/s and the other one
  is moving at v2 3.88 m/s, calculate
• the velocity of the lighter car after the
  collision.
• the velocity of the heavier car after the
  collision.
• the change in momentum of the lighter car.
• the change in momentum of the heavier car.

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Elastic Collision in the CM CAPA 6-9
3.88 m/s

• Calcuate Vcm (M1V1bM2V2b)/(M1M2)
• Calcuate U1b and U2b
• U1b V1b Vcm and U2b V2b Vcm
• Set U1a -U1b and U2a -U2b
• Calcuate V1a and V2a
• V1a U1a Vcm and V2a U2a Vcm

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Elastic Collision in the CM CAPA 6-9
U2b
U1b
U1a
U2a

• Calcuate Vcm (M1V1bM2V2b)/(M1M2)
• Calcuate U1b and U2b
• U1b V1b Vcm and U2b V2b Vcm

Vcm (M1V1bM2V2b)/(M1M2) Vcm (4674.23
5673.88)/(467567) 4.038 m/s U1b V1a - Vcm
4.23 m/s - 4.038 m/s 0.192 m/s U2b V2b -
Vcm 3.88 m/s - 4.038 m/s -0.158 m/s
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Elastic Collision in the CM CAPA 6-9
U2b
U1b
U1a
U2a

• Set U1a -U1b and U2a -U2b
• Calcuate V1a and V2a
• V1a U1a Vcm and V2a U2a Vcm

U1a - U1b -0.192 m/s U2a -U2b 0.158
m/s V1a U1a Vcm -0.192 m/s 4.038 m/s
3.846 m/s V2a U2a Vcm 0.158 m/s 4.038
m/s 4.196 m/s
3.85 m/s
4.20 m/s
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Elastic Collision in the CM CAPA 6-9
Impulse mVa mVb (for each of cars 1 and 2)
Note Impulse on car 1 - Impulse on car 2.
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Next Time

• Chapter 7 Conservation of Momentum.
• Quiz on Chapter 7.
• Please see me with any questions or comments.

See you on Monday.