Impulse and Momentum.pdf
Transcript of Impulse and Momentum.pdf
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Chapter 9. Impulse andChapter 9. Impulse and
MomentumMomentumExplosions and collisionsobey some surprisingly
simple laws that makeproblem solving easier whencomparing the situation
beforeand afteraninteraction.
Chapter Goal: To introduce
the ideas of impulse andmomentum and to learn anew problem-solving strategy
based on conservation laws.
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Topics:
Momentum and Impulse Solving Impulse and Momentum
Problems
Chapter 9. Impulse andChapter 9. Impulse andMomentumMomentum
Conservation of Momentum Inelastic Collisions
Explosions
Momentum in Two Dimensions
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Momentum
A B10 m/s
After the collisionA B
v u
3
What is the velocity of ball A after the collision? ball B?
What is conserved during the collision?
MOMENTUM
The total momentum is the sum of momentum of ball A and
momentum of ball B.
p mv====r rr rr rr r
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Momentum
The total momentum of the system is conserved during the collision:
A B10 m/s
AB
p mv====r rr rr rr r
,A A i A Bv m v m u= += += += +
4
Momentum is a vector. It has the same direction as corresponding
velocity.
General expression for the momentum conservation: the total
momentum before the collision is equal to the total momentum after
the collision
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Momentum
A
B
A
p mv====r rr rr rr r
General expression for the momentum conservation: the totalmomentum before the collision is equal to the total momentum after
the collision
,ivrrrr
,B iv
rrrr
,A fvrrrr
5
B
, , , ,A A i B B i A A f B B f
v m v m v m v+ = ++ = ++ = ++ = +r r r r r r r r r r r r r r r r
,B fv
Usually this equation is written in terms of components.
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Example:
A B10 m/s
After the collision the balls are moving together(have the same velocity). What is their velocity?
1A
m kg==== 4Bm kg====
6
v
, 10
i A A i
g mp m v
s
= == == == =Momentum before the collision:
Momentum after the collision: ( ) 5f A Bp m m v v= + == + == + == + =
Conservation of momentum: i fp p====
10 5v==== 2 /v m s====
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i ixp mv====
0f fx
p mv=
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2
1
1 ,1 1 ,1 ,2 1
t
fx ix x on
t
m v m v F dt = = = =
2
1
2 ,2 2 ,2 ,1 2
t
fx ix x on
t
m v m v F dt = = = =
Newtons third law:
F F= = = =
9
Then
2 2
1 1
,1 2 ,2 1
t t
x on x ont t
F dt F dt= = = =
(((( ))))2 ,2 2 ,2 1 ,1 1 ,1
fx ix fx ix
m v m v m v m v = = = =
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(((( ))))2 ,2 2 ,2 1 ,1 1 ,1fx ix fx ixm v m v m v m v = = = =
1 ,1 2 ,2 1 ,1 2 ,2ix ix fx fxm v m v m v m v+ = ++ = ++ = ++ = +
====
10
, , , ,x x x x
, ,ix total fx total p p====
The law of conservation of momentum
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Momentum
A A
p mv====r rr rr rr r
rrrrrrrr
The law of conservation of momentum:
The total momentum of an isolated system (no external forces) does notchange.
Interactions within system do not change the systems total momentum
isolated system
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B
B
, , , ,A A i B B i A A f B B fv m v m v m v+ = ++ = ++ = ++ = +
r r r r r r r r r r r r r r r r
,
,iv
rrrr
,A f
,B fv
rrrr
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Momentum p mv====r rr rr rr r
The ball is dropped onto a hard floor:
The ball is not an isolated system (interaction with the floor)
no conservation of momentum for the ball
Initial momentum is
Final momentum (after collision) is
i ip mv====
r rr rr rr r
f fp mv====
r rr rr rr r
12
iv
rrrr
vrrrr
The ball+ the floor is an isolated system
The total momentum (ball+floor) is conserved
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Example: Find2x
v
13
Motion with constantacceleration:
2
1 , 1( ) 2 16
x B xv a x= == == == =
1 , 4 /
x Bv m s====
Isolated system
Momentum before the collision:
, 1 , 1 , 1 , 75 4 300
i total B x B C x C B x B
g mp m v m v m v
s
= + = = == + = = == + = = == + = = =
Momentum after the collision:
, 2 , 2 , 2 2( ) 100
total B x B C x C B C x xp m v m v m m v v= + = + == + = + == + = + == + = + =
Conservation of momentum:
, ,f total i totalp p====
2100 300
xv ==== 2 3 /xv m s====
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Perfectly inelastic collision:
A collision in which the two objects stick together and move with a
common final velocity.
, 1 ,1 2 ,2i total ix ixp m v m v= += += += +
14
, 1 2
( )total fx
p m m v= += += += +
1 ,1 2 ,2 1 2( )
ix ix fxm v m v m m v+ = ++ = ++ = ++ = +
1 1
,2 ,1
2 2
3 3
1 0.5 2.25 / 2 2ix fx ix
m mv v v m sm m
= + = = = + = = = + = = = + = =
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General Principles
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General Principles
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Important Concepts
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Important Concepts
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Applications
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Applications
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The carts change of momentum is
A. 30 kg m/s.B. 10 kg m/s.
C.10 kg m/s.
D.20 kg m/s.E.30 kg m/s.
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The carts change of momentum is
A. 30 kg m/s.B. 10 kg m/s.
C.10 kg m/s.
D.20 kg m/s.E.30 kg m/s.
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A. They exert equal impulses because they
A 10 g rubber ball and a 10 g clay ball arethrown at a wall with equal speeds. Therubber ball bounces, the clay ball sticks.
Which ball exerts a larger impulse on thewall?
ave equa momen a.B. The clay ball exerts a larger
impulse because it sticks.C. Neither exerts an impulse on the
wall because the wall doesnt move.D. The rubber ball exerts a larger impulse
because it bounces.
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A. They exert equal impulses because they
A 10 g rubber ball and a 10 g clay ball arethrown at a wall with equal speeds. Therubber ball bounces, the clay ball sticks.
Which ball exerts a larger impulse on thewall?
.B. The clay ball exerts a larger
impulse because it sticks.C. Neither exerts an impulse on the
wall because the wall doesnt move.D. The rubber ball exerts a larger impulse
because it bounces.
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Objects A and C are made of different materials, with differentspringiness, but they have the same mass and are initially at rest.When ball B collides with object A, the ball ends up at rest. When ball B
is thrown with the same speed and collides with object C, the ball
rebounds to the left. Compare the velocities of A and C after thecollisions. Is vA greater than, equal to, or less than vC?
A. vA > vCB. vA < vCC. vA = vC
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Objects A and C are made of different materials, with differentspringiness, but they have the same mass and are initially at rest.When ball B collides with object A, the ball ends up at rest. When ball B
is thrown with the same speed and collides with object C, the ball
rebounds to the left. Compare the velocities of A and C after thecollisions. Is vA greater than, equal to, or less than vC?
A. vA > vCB. vA < vCC. vA = vC
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The two particles are both moving to the right. Particle 1 catches
up with particle 2 and collides with it. The particles stick togetherand continue on with velocity vf. Which of these statements istrue?
A. vf = v2.
B. vf is less than v2.
C. vf
is greater than v2
, but less than v1
.
D. vf = v1.
E. vf is greater than v1.
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The two particles are both moving to the right. Particle 1 catches
up with particle 2 and collides with it. The particles stick togetherand continue on with velocity vf. Which of these statements istrue?
A. vf = v2.
B. vf is less than v2.
C. vf
is greater than v2
, but less than v1
.
D. vf = v1.
E. vf is greater than v1.
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An explosion in a rigid pipe shoots outthree pieces. A 6 g piece comes out theright end. A 4 g piece comes out the left
end with twice the speed of the 6 gpiece. From which end does the third
iece emer e?
A. Right end
B. Left end
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An explosion in a rigid pipe shoots outthree pieces. A 6 g piece comes out theright end. A 4 g piece comes out the left
end with twice the speed of the 6 gpiece. From which end does the third
iece emer e?
A. Right end
B. Left end