10/27/09 1 Chapter 8 Conservation of Linear Momentum Physics 201 October 22, 2009 Conservation of Linear Momentum • Definition of linear momentum, ! p ! p = m ! v Linear momentum is a vector. Units of linear momentum are kg-m/s. Can write Newton’s second law in terms of momentum: d ! p dt = d(m ! v) dt = m d ! v dt = m ! a ! d ! p dt = ! F net
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10/27/09
1
Chapter 8
Conservation of Linear Momentum
Physics 201
October 22, 2009
Conservation of Linear Momentum
•! Definition of linear momentum,
! p
! p = m
! v
Linear momentum is a vector.
Units of linear momentum are kg-m/s.
Can write Newton’s second law in terms of momentum:
d! p
dt=
d(m! v )
dt= m
d! v
dt= m! a
!d! p
dt=! F net
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Momentum of a system of particles
•! The total momentum of a system of
particles is the vector sum of the momenta of
the individual particles:
From Newton’s second law, we obtain
Psys
! "!!= m
i
"vi
i
! ="pi
i
!
!Fext
i
! =!Fnetext =
d!Psys
dti
!
Conservation of Momentum
•! Law of conservation of momentum:
–! If the sum of the external forces on a system is
zero, the total momentum of the system does not
change.
If then
Momentum is always conserved (even if forces are nonconservative).
!Psys = mi
!vi
i
! = M!vCM = const
" !""""
!Fext
i
! = 0
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Collisions
“before” m1 m2
“after” m1 m2
momentum before collision = momentum after collision
Always -
But only if
! F
external= 0
Explosion - I
“before” M
“after” m1 m2
v1 v2
Example: m1 = M/3 m2 = 2M/3
After explosion, which block has larger momentum? (left, right, same)
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“before” M
“after” m1 m2
v1 v2
Example: m1 = M/3 m2 = 2M/3
After explosion, which block has larger momentum? (left, right, same)
Each has the same momentum
Explosion - I
“before” M
“after” m1 m2
v1 v2
Example: m1 = M/3 m2 = 2M/3
After explosion, which block has larger momentum? (left, right, same)
Each has the same momentum
Which block has larger velocity?
Explosion - I
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“before” M
“after” m1 m2
v1 v2
Example: m1 = M/3 m2 = 2M/3
After explosion, which block has larger momentum? (left, right, same)
Each has the same momentum
Which block has larger velocity?
mv is the same for each block, so smaller mass has larger velocity
Explosion - I
“before” M
“after” m1 m2
v1 v2
Example: m1 = M/3 m2 = 2M/3
After explosion, which block has larger momentum? (left, right, same)
Each has the same momentum
Which block has larger velocity?
mv is the same for each block, so smaller mass has larger velocity
Is kinetic energy conserved?
Explosion - I
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Explosion - I
“before” M
“after” m1 m2
v1 v2
Example: m1 = M/3 m2 = 2M/3
After explosion, which block has larger momentum? (left, right, same)
Each has the same momentum
Which block has larger velocity?
mv is the same for each block, so smaller mass has larger velocity
Is kinetic energy conserved? NO! K was 0 before, it is greater after the explosion.
Explosion - I
“before” M
“after” m1 m2
v1 v2
Example: m1 = M/3 m2 = 2M/3
After explosion, which block has larger momentum? (left, right, same)
Each has the same momentum
Which block has larger velocity?
mv is the same for each block, so smaller mass has larger velocity
Is kinetic energy conserved? (green=yes, red=no) NO!
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This is like a microscopic explosion ….
Momentum and Impulse
!Fave!t " I definition of impulse
!F = m
!a = m
d!v
dt=d!p
dt# !!p =!F!t
!p ! m
!v
!! For single object….
"! If F = 0, then momentum conserved (p = 0)
!psys
=!pi
i
!
Internal forces " forces between objects in system
External forces " any other forces
#!psys
=!Fext#t
Thus, if !Fext
= 0, then #!psys
= 0
i.e. total momentum is conserved!
•!For “system” of objects …
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Momentum and Impulse
!Fave!t " I definition of impulse
!F = m
!a = m
d!v
dt=d!p
dt# !!p =!F!t
!p ! m
!v
!! For single object….
"! If F = 0, then momentum conserved (p = 0)
!psys
=!pi
i
!
Internal forces " forces between objects in system
External forces " any other forces
#!psys
=!Fext#t
Thus, if !Fext
= 0, then #!psys
= 0
i.e. total momentum is conserved!
•!For “system” of objects …
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!Fave!t " I definition of impulse
!F = m
!a = m
d!v
dt=d!p
dt# !!p =!F!t
Let’s estimate the average force during the collision
Club speed: 50 m/s
Assume that impulse is given after 5 cm
--> whiteboard
!Fave =
I
!t=1
!t
!Fdt
ti
t f
"
Some Terminology
•! Elastic Collisions:
collisions that conserve kinetic energy
•! Inelastic Collisions:
collisions that do not conserve kinetic energy
*! Completely Inelastic Collisons:
objects stick together
n.b. ALL CONSERVE MOMENTUM!!
If external forces = 0
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Elastic Collision in 1-Dimension
m1v1i + m2
v2i = m1
v1 f + m2
v2 f
1
2m1v1i
2+1
2m2v2i
2=1
2m1v1 f
2+1
2m2v2 f
2
Energy conserved (for elastic
collision only)
Linear momentum is conserved
Initial Final
Elastic Collision Conservation of Momentum
m1v1i + m2v2i = m1v1 f + m2v2 f
m1(v1i ! v1 f ) = m2 (v2 f ! v2i )
Conservation of Kinetic Energy
1
2m1v1i
2+
1
2m2v2i
2=
1
2m1v1 f
2+
1
2m2v2 f
2
m1(v1i
2! v1 f
2 ) = m2 (v2 f
2! v2i
2 )
m1(v1i ! v1 f )(v1i + v1 f ) = m2 (v2 f ! v2i )(v2 f + v2i )
Combining the above two equations
v1i + v1 f = v2i + v2 f
v1i ! v2i = !(v1 f ! v2 f )
Magnitude of relative velocity is conserved.
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Is this an elastic collision?
v1i ! v2i = !(v
1 f ! v2 f )For elastic collision only:
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What is the speed of the golf ball, in
case of an elastic collision
Club speed: 50 m/s
Mass of clubhead: 0.5kg
Mass of golfball: 0.05kg
Two unknowns:
speed of club and
speed of golfball after impact
Problem solving strategy:
-! Momentum conservation
-! Energy conservation (or
use the derived equation
for relative velocities)
--> whiteboard
Is this an elastic collision?
v1i ! v2i = !(v
1 f ! v2 f )Yes, the relative speeds
are approximately the same
before and after collision
For elastic collision only:
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v1 f =
m1! m
2
m1+ m
2
v1i
v2 f =
2m1
m1+ m
2
v1i
Result:
Special cases: 1)! Golf shot: m1>>m2
Club speed almost unchanged
Ball speed almost 2 x club speed
2) Neutron scatters on heavy nucleus: m1<<m2
neutron scatters back with almost same speed speed of nucleus almost unchanged
Completely inelastic collision
•! Two objects stick together and move with the center
of mass:
•! If pAi=0:
!PAi +
!PBi =
!PAf +
!PBf =
!PCM
!PBi =
!PCM
mBv!
Bi = mA + mB( )v!
f
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!! Two stage process:
1. m collides with M, inelastically. Both M and m then move together with a velocity V
f
(before having risen significantly).
2. Both (m1 + m2) rise a height h, conserving energy E. (no non-conservative forces acting after collision)
Ballistic
Pendulum
What is the initial
velocity vli of the
projectile? Known quantities:
m1, m2, h
•! Stage 1: Momentum is conserved
Energy is not conserved
in x-direction:
!! Stage 2 (after the collision): Energy is conserved