momentum and momentum conservation

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March 24, 2009 Momentum and Momentum Conservation Momentum Impulse Conservation of Momentum Collision in 1-D Collision in 2-D

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Momentum and Momentum Conservation. Momentum Impulse Conservation of Momentum Collision in 1-D Collision in 2-D. Linear Momentum. A new fundamental quantity, like force, energy - PowerPoint PPT Presentation

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Page 1: Momentum and  Momentum Conservation

March 24, 2009

Momentum and Momentum Conservation

Momentum Impulse Conservation of Momentum Collision in 1-D Collision in 2-D

Page 2: Momentum and  Momentum Conservation

March 24, 2009

Linear Momentum A new fundamental quantity, like force,

energy The linear momentum p of an object of

mass m moving with a velocity is defined to be the product of the mass and velocity:

The terms momentum and linear momentum will be used interchangeably in the text

Momentum depend on an object’s mass and velocity

v

vmp

Page 3: Momentum and  Momentum Conservation

March 24, 2009

Linear Momentum, cont Linear momentum is a vector quantity

Its direction is the same as the direction of the velocity

The dimensions of momentum are ML/T The SI units of momentum are kg · m / s Momentum can be expressed in

component form: px = mvx py = mvy pz = mvz

mp v

Page 4: Momentum and  Momentum Conservation

March 24, 2009

Newton’s Law and Momentum

Newton’s Second Law can be used to relate the momentum of an object to the resultant force acting on it

The change in an object’s momentum divided by the elapsed time equals the constant net force acting on the object

tvm

tvmamFnet

)(

netFtp

interval timemomentumin change

Page 5: Momentum and  Momentum Conservation

March 24, 2009

Impulse When a single, constant force acts on the

object, there is an impulse delivered to the object is defined as the impulse The equality is true even if the force is not

constant Vector quantity, the direction is the same as

the direction of the force

I

tFI

netFtp

interval timemomentumin change

Page 6: Momentum and  Momentum Conservation

March 24, 2009

Impulse-Momentum Theorem

The theorem states that the impulse acting on a system is equal to the change in momentum of the system

if vmvmpI

ItFp net

Page 7: Momentum and  Momentum Conservation

March 24, 2009

Calculating the Change of Momentum

0 ( )p m v mv

( )

after before

after before

after before

p p p

mv mv

m v v

For the teddy bear

For the bouncing ball

( ) 2p m v v mv

Page 8: Momentum and  Momentum Conservation

March 24, 2009

How Good Are the Bumpers? In a crash test, a car of mass 1.5103 kg collides with a wall

and rebounds as in figure. The initial and final velocities of the car are vi=-15 m/s and vf = 2.6 m/s, respectively. If the collision lasts for 0.15 s, find

(a) the impulse delivered to the car due to the collision (b) the size and direction of the average force exerted on the car

Page 9: Momentum and  Momentum Conservation

March 24, 2009

How Good Are the Bumpers? In a crash test, a car of mass 1.5103 kg collides with a wall

and rebounds as in figure. The initial and final velocities of the car are vi=-15 m/s and vf = 2.6 m/s, respectively. If the collision lasts for 0.15 s, find

(a) the impulse delivered to the car due to the collision (b) the size and direction of the average force exerted on the car

smkgsmkgmvp ii /1025.2)/15)(105.1( 43

Ns

smkgt

ItpFav

54

1076.115.0

/1064.2

smkgsmkgmvp ff /1039.0)/6.2)(105.1( 43

smkg

smkgsmkg

mvmvppI ifif

/1064.2

)/1025.2()/1039.0(4

44

Page 10: Momentum and  Momentum Conservation

March 24, 2009

Conservation of Momentum In an isolated and closed

system, the total momentum of the system remains constant in time. Isolated system: no external

forces Closed system: no mass

enters or leaves The linear momentum of each

colliding body may change The total momentum P of the

system cannot change.

Page 11: Momentum and  Momentum Conservation

March 24, 2009

Conservation of Momentum Start from impulse-

momentum theorem

Since

Then

So

if vmvmtF 222212

if vmvmtF 111121

tFtF 1221

)( 22221111 ifif vmvmvmvm

ffii vmvmvmvm 22112211

Page 12: Momentum and  Momentum Conservation

March 24, 2009

Conservation of Momentum When no external forces act on a system

consisting of two objects that collide with each other, the total momentum of the system remains constant in time

When then For an isolated system

Specifically, the total momentum before the collision will equal the total momentum after the collision

ifnet ppptF

0p

0netF

if pp

ffii vmvmvmvm 22112211

Page 13: Momentum and  Momentum Conservation

March 24, 2009

The Archer An archer stands at rest on frictionless ice and fires a 0.5-kg

arrow horizontally at 50.0 m/s. The combined mass of the archer and bow is 60.0 kg. With what velocity does the archer move across the ice after firing the arrow?

ffii vmvmvmvm 22112211

fi pp

?,/50,0,5.0,0.60 122121 ffii vsmvvvkgmkgm

ff vmvm 22110

smsmkg

kgvmmv ff /417.0)/0.50(

0.605.0

21

21

Page 14: Momentum and  Momentum Conservation

March 24, 2009

Types of Collisions Momentum is conserved in any collision Inelastic collisions: rubber ball and hard

ball Kinetic energy is not conserved Perfectly inelastic collisions occur when the

objects stick together Elastic collisions: billiard ball

both momentum and kinetic energy are conserved

Actual collisions Most collisions fall between elastic and

perfectly inelastic collisions

Page 15: Momentum and  Momentum Conservation

Collision between two objects of the same mass. One mass is at rest.

Collision between two objects. One not at rest initially has twice the mass.

Collision between two objects. One at rest initially has twice the mass.

Simple Examples of Head-On Collisions(Energy and Momentum are Both Conserved)

vmp

Page 16: Momentum and  Momentum Conservation

vmp

Collision between two objects of the same mass. One mass is at rest.

Example of Non-Head-On Collisions

(Energy and Momentum are Both Conserved)

If you vector add the total momentum after collision,you get the total momentum before collision.

Page 17: Momentum and  Momentum Conservation

March 24, 2009

Collisions Summary In an elastic collision, both momentum and

kinetic energy are conserved In an inelastic collision, momentum is conserved

but kinetic energy is not. Moreover, the objects do not stick together

In a perfectly inelastic collision, momentum is conserved, kinetic energy is not, and the two objects stick together after the collision, so their final velocities are the same

Elastic and perfectly inelastic collisions are limiting cases, most actual collisions fall in between these two types

Momentum is conserved in all collisions

Page 18: Momentum and  Momentum Conservation

March 24, 2009

More about Perfectly Inelastic Collisions

When two objects stick together after the collision, they have undergone a perfectly inelastic collision

Conservation of momentum

Kinetic energy is NOT conserved

fii vmmvmvm )( 212211

21

2211

mmvmvmv ii

f

Page 19: Momentum and  Momentum Conservation

March 24, 2009

An SUV Versus a Compact An SUV with mass 1.80103 kg is travelling

eastbound at +15.0 m/s, while a compact car with mass 9.00102 kg is travelling westbound at -15.0 m/s. The cars collide head-on, becoming entangled.

(a) Find the speed of the entangled cars after the collision.

(b) Find the change in the velocity of each car.

(c) Find the change in the kinetic energy of the system consisting of both cars.

Page 20: Momentum and  Momentum Conservation

March 24, 2009

(a) Find the speed of the entangled cars after the collision.

fii vmmvmvm )( 212211

fi pp smvkgm

smvkgm

i

i

/15,1000.9

/15,1080.1

22

2

13

1

21

2211

mmvmvmv ii

f

smv f /00.5

An SUV Versus a Compact

Page 21: Momentum and  Momentum Conservation

March 24, 2009

(b) Find the change in the velocity of each car.

smvvv if /0.1011

smvkgm

smvkgm

i

i

/15,1000.9

/15,1080.1

22

2

13

1

smv f /00.5

An SUV Versus a Compact

smvvv if /0.2022

smkgvvmvm if /108.1)( 41111

02211 vmvm

smkgvvmvm if /108.1)( 42222

Page 22: Momentum and  Momentum Conservation

March 24, 2009

(c) Find the change in the kinetic energy of the system consisting of both cars.

JvmvmKE iii52

22211 1004.3

21

21

smvkgm

smvkgm

i

i

/15,1000.9

/15,1080.1

22

2

13

1

smv f /00.5

An SUV Versus a Compact

JKEKEKE if51070.2

JvmvmKE fff42

22211 1038.3

21

21

Page 23: Momentum and  Momentum Conservation

March 24, 2009

More About Elastic Collisions Both momentum and kinetic

energy are conserved

Typically have two unknowns Momentum is a vector quantity

Direction is important Be sure to have the correct signs

Solve the equations simultaneously

222

211

222

211

22112211

21

21

21

21

ffii

ffii

vmvmvmvm

vmvmvmvm

Page 24: Momentum and  Momentum Conservation

March 24, 2009

Elastic Collisions A simpler equation can be used in place of the

KE equation

iffi vvvv 2211

)vv(vv f2f1i2i1

222

211

222

211 2

121

21

21

ffii vmvmvmvm

))(())(( 2222211111 ififfifi vvvvmvvvvm

)()( 222111 iffi vvmvvm

)()( 22

222

21

211 iffi vvmvvm

ffii vmvmvmvm 22112211

ffii vmvmvmvm 22112211

Page 25: Momentum and  Momentum Conservation

March 24, 2009

Summary of Types of Collisions

In an elastic collision, both momentum and kinetic energy are conserved

In an inelastic collision, momentum is conserved but kinetic energy is not

In a perfectly inelastic collision, momentum is conserved, kinetic energy is not, and the two objects stick together after the collision, so their final velocities are the same

iffi vvvv 2211 ffii vmvmvmvm 22112211

ffii vmvmvmvm 22112211

fii vmmvmvm )( 212211

Page 26: Momentum and  Momentum Conservation

March 24, 2009

Problem Solving for 1D Collisions, 1

Coordinates: Set up a coordinate axis and define the velocities with respect to this axis It is convenient to make

your axis coincide with one of the initial velocities

Diagram: In your sketch, draw all the velocity vectors and label the velocities and the masses

Page 27: Momentum and  Momentum Conservation

March 24, 2009

Problem Solving for 1D Collisions, 2

Conservation of Momentum: Write a general expression for the total momentum of the system before and after the collision Equate the two total

momentum expressions Fill in the known values

ffii vmvmvmvm 22112211

Page 28: Momentum and  Momentum Conservation

March 24, 2009

Problem Solving for 1D Collisions, 3

Conservation of Energy: If the collision is elastic, write a second equation for conservation of KE, or the alternative equation This only applies to

perfectly elastic collisions

Solve: the resulting equations simultaneously

iffi vvvv 2211

Page 29: Momentum and  Momentum Conservation

March 24, 2009

One-Dimension vs Two-Dimension

Page 30: Momentum and  Momentum Conservation

March 24, 2009

Two-Dimensional Collisions For a general collision of two objects in two-

dimensional space, the conservation of momentum principle implies that the total momentum of the system in each direction is conserved

fyfyiyiy

fxfxixix

vmvmvmvm

vmvmvmvm

22112211

22112211

Page 31: Momentum and  Momentum Conservation

March 24, 2009

Two-Dimensional Collisions The momentum is conserved in all

directions Use subscripts for

Identifying the object Indicating initial or final values The velocity components

If the collision is elastic, use conservation of kinetic energy as a second equation Remember, the simpler equation can only be

used for one-dimensional situations

fyfyiyiy

fxfxixix

vmvmvmvm

vmvmvmvm

22112211

22112211

iffi vvvv 2211

Page 32: Momentum and  Momentum Conservation

March 24, 2009

Glancing Collisions

The “after” velocities have x and y components

Momentum is conserved in the x direction and in the y direction

Apply conservation of momentum separately to each direction

fyfyiyiy

fxfxixix

vmvmvmvm

vmvmvmvm

22112211

22112211

Page 33: Momentum and  Momentum Conservation

March 24, 2009

2-D Collision, example Particle 1 is moving

at velocity and particle 2 is at rest

In the x-direction, the initial momentum is m1v1i

In the y-direction, the initial momentum is 0

1iv

Page 34: Momentum and  Momentum Conservation

March 24, 2009

2-D Collision, example cont After the collision, the

momentum in the x-direction is m1v1f cos q m2v2f cos f

After the collision, the momentum in the y-direction is m1v1f sin q m2v2f sin f

If the collision is elastic, apply the kinetic energy equation

fq

fq

sinsin00

coscos0

2211

221111

ff

ffi

vmvm

vmvmvm

222

211

211 2

121

21

ffi vmvmvm

Page 35: Momentum and  Momentum Conservation

March 24, 2009

Problem Solving for Two-Dimensional Collisions

Coordinates: Set up coordinate axes and define your velocities with respect to these axes It is convenient to choose the x- or y-

axis to coincide with one of the initial velocities

Draw: In your sketch, draw and label all the velocities and masses

Page 36: Momentum and  Momentum Conservation

March 24, 2009

Problem Solving for Two-Dimensional Collisions, 2

Conservation of Momentum: Write expressions for the x and y components of the momentum of each object before and after the collision

Write expressions for the total momentum before and after the collision in the x-direction and in the y-direction

fyfyiyiy

fxfxixix

vmvmvmvm

vmvmvmvm

22112211

22112211

Page 37: Momentum and  Momentum Conservation

March 24, 2009

Problem Solving for Two-Dimensional Collisions, 3

Conservation of Energy: If the collision is elastic, write an expression for the total energy before and after the collision Equate the two expressions Fill in the known values Solve the quadratic equations

Can’t be simplified

Page 38: Momentum and  Momentum Conservation

March 24, 2009

Problem Solving for Two-Dimensional Collisions, 4

Solve for the unknown quantities Solve the equations simultaneously There will be two equations for

inelastic collisions There will be three equations for

elastic collisionsCheck to see if your answers are

consistent with the mental and pictorial representations. Check to be sure your results are realistic

Page 39: Momentum and  Momentum Conservation

March 24, 2009

Collision at an Intersection A car with mass 1.5×103 kg

traveling east at a speed of 25 m/s collides at an intersection with a 2.5×103 kg van traveling north at a speed of 20 m/s. Find the magnitude and direction of the velocity of the wreckage after the collision, assuming that the vehicles undergo a perfectly inelastic collision and assuming that friction between the vehicles and the road can be neglected. ??,/20,/25

105.2,105.1 33

qfviycix

vc

vsmvsmvkgmkgm

Page 40: Momentum and  Momentum Conservation

March 24, 2009

Collision at an Intersection

??,/20,/25105.2,105.1 33

qfviycix

vc

vsmvsmvkgmkgm

smkgvmvmvmp cixcvixvcixcxi /1075.3 4

qcos)( fvcvfxvcfxcxf vmmvmvmp

qcos)1000.4(/1075.3 34fvkgsmkg

smkgvmvmvmp viyvviyvciycyi /1000.5 4

qsin)( fvcvfyvcfycyf vmmvmvmp

qsin)1000.4(/1000.5 34fvkgsmkg

Page 41: Momentum and  Momentum Conservation

March 24, 2009

Collision at an Intersection

??,/20,/25105.2,105.1 33

qfviycix

vc

vsmvsmvkgmkgm

qcos)1000.4(/1075.3 34fvkgsmkg

qsin)1000.4(/1000.5 34fvkgsmkg

33.1/1075.3/1000.5tan 4

4

smkgsmkgq

1.53)33.1(tan 1 q

smkg

smkgv f /6.151.53sin)1000.4(

/1000.53

4