201: Physics Lecture 14
q q q q
Momentum Impulse Collisions: Elastic and Inelastic Conservation of Momentum and Energy
05/14/09
Physics 201, UW-Madison
1
Momentum and Impulse
q
Momentum
r p
r Fave =
= = =
q
For single object....
If F = 0, then momentum conserved (p = 0)
For "system" of objects ... r p =
tot
= = 0, = 0 ,
.. !
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Collisions
Procedure
m 1 m 2 "be " fore Draw "before", "after" Define system so that Fext = 0 m 1 m 2 "afte r" Set up axes Compute Ptotal "before" Compute Ptotal "after"
Explosions
M "be " fore
Set them equal to each other
m 1
05/14/09
m 2
"afte r"
Physics 201, UW-Madison
3
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!!
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Elastic Collision in 1-Dimension
QuickTime and aCinepak decompressorare needed to see this picture.
m1v1i + m2v 2i = m1v1 f + m2v 2 f 1 1 1 1 2 2 2 m1v1i + m 2v 2i = m1v12f + m 2v 2 f 2 2 2 2
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Elastic Collision
Conservation of Momentum m1v1i + 2 = 1 + 2 2 1 2
1 ( - ) = 2 ( - ) 1 1 2 2
1 1 1 1 2 2 2 2 1 + 2 = 1 + 2 1 2 1 2 2 2 2 2 2 2 2 2 1 ( - ) = 2 ( - ) 1 1 2 2
1 ( - )( + ) = 2 ( - )( + ) 1 1 1 1 2 2 2 2
+ = + 1 1 2 2
- = -( - ) 1 2 1 2
Magnitudeof re lativeve locity is conse d. rve
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Question
A railroad car is coasting along a horizontal track with spe d V whe it runs into and conne e n cts with a se cond ide ntical railroad car, initially at re Assum the is no friction be e the st. ing re twe n cars and therails, what is thespe d of thetwo couple cars afte thecollision? e d r
A. V B. V/2 C. V/4 D. 0
pA,i + 0 = + , :
C ORREC T
+ = 2 , = 2 + , + , = , 2
05/14/09
Physics 201, UW-Madison
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Question part B
What physical quantities are conserved in the above collision? A. Only momentum is conserved B. Only total mechanical energy is conserved C. Both are conserved D. Neither are conserved C ORREC T
Momentum is conserved because the net external force is 0. Because the cars stay coupled, the collision is inelastic and kinetic energy is NOT conserved
05/14/09
Physics 201, UW-Madison
8
Explosions - I
M "be " fore
v1
v2 m 1 m 2
"afte r"
Which block has larger momentum, green, blue (same=yellow)? Each has same momentum Which block has larger velocity? mv same for each --- smaller mass has larger velocity
Which block has larger kinetic energy? KE = mv2/2 = m2v2/2m = p2/2m --- mass smaller has larger KE Example: m = M/3 m = 2M/3 *NO!! Is kinetic energy conserved?
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Collisions or Explosions in Two Dimensions
y
x
be fore
afte r
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Physics 201, UW-Madison
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Explosions
M "be " fore
A
B
Which of these is possible? A B both
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Explosions
M "be " fore
A
B
Which of the is possible se ? A B ne r ithe
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Ballistic Pendulum
L L m v M L H M+m V L
V=0
A projectile of mass m moving horizontally with speed v strikes a stationary mass M suspended by strings of length L. Subsequently, m + M rise to a height of H. Given H, what is the initial speed v of the projectile?
05/14/09 Physics 201, UW-Madison
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Ballistic Pendulum
q
Two stage process:
1. m collides with M, inelastically. Both M and m then move together with a velocity V (before having risen significantly). 2. M and m rise a height H, conserving energy E. (no non-conservative forces acting after collision)
05/14/09
Physics 201, UW-Madison
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Ballistic Pendulum
q
Stage 1: Momentum is conserved
Energy is not conserved
in x-direction:
mv = ( + )
q
Stage 2: Energy is conserved
1 K + = ( + ) 2 + ( + ) 2 1 = 2 ( + ) 2 = ( + ) 2
Substituting for V gives:
v = 1+
2
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05/14/09
Physics 201, UW-Madison
Ballistic Pendulum
L H M+m d
Its easier to measure d than to measure H, so after a small bit of geometry we get:
L
d = :
2 = (1 - ) : + ... 2
v = 1+
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Physics 201, UW-Madison
Shooting Pool...
q
Assuming Collision is elastic (KE is conserved) No spin is imparted Balls have the same mass One ball starts out at rest Then the angle between the ba...