Momentum & Impulse

Level 1 Physics

What you need to know Essential Questions

How is impulse and momentum related?

How does the law of conservation of momentum apply to objects at rest? In motion?

Objectives Define and give examples of

impulse and momentum Restate Newton’s Second Law in

terms of momentum Calculate the change in momentum

from the area under the curve of a force versus time graph

Derive a statement of the conservation of momentum between two objects by applying Newton’s Third Law

Define and recognize examples of elastic and inelastic collisions

Explain which conservation laws apply to each type of collision

Demonstrate proficiency in solving problems involving conservation of momentum in collisions

Impulse Equals Change in Momentum

Remember the following; N.S.L &definition of acceleration

€

ΣFm= a, a =

Δv

ΔtΣF

m=Δv

Δt⇒ ΣFΔt = mΔv

ΣFΔt→Im pulse(J)

mΔv→Change in momentum (Δp)

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Im pulse =Change in momentum

J = Δp

ΣFΔt = mΔv

Units of impulse: N*sUnits of momentum: kg*m/s

Impulse and Momentum

€

FΔt = mΔvImpulse

Change in Momentum

Notice the relationshipwith force and time

€

F =mΔv

Δt

Impulse – Momentum Relationships

Impulse – Momentum Relationships

VmfT Constant

Since TIME is directly related to the VELOCITY when the force and mass are constant, the LONGER the cannonball is in the barrel the greater the velocity.

Also, you could say that the force acts over a larger displacement, thus there is more WORK. The work done on the cannonball turns into kinetic energy.

Impulse – Momentum Relationships

Which would be more damaging: driving into a massive concrete wall, or driving at the same speed into a head-on collision with an identical car traveling toward you at the same speed?

Both cases are equivalent, because either way, your car is rapidly decelerating to a dead stop. The dead stop is easy to see when hitting the wall, and a little thought will show the same is true when hitting the car. If the oncoming car were traveling slower, with less momentum, you'd keep going after the collision with more 'give,' and less damage (to you!). But if the oncoming car had more momentum than you, it would keep going and you'd snap into a sudden reverse with greater damage. Identical cars at equal speeds means equal momenta -- zero before, zero after collision.

CollisionsConsider 2 objects heading towardsone another. Upon colliding…

N.T.L. says that the FORCE theyexert on each other is EQUAL but in the OPPOSITE direction

The time of impact is the SAME for both…

Therefore, the IMPULSES of the 2 objects are also EQUAL and OPPOSITE

€

F1 = −F2 t1 = t2Ft( )1 = −Ft( )2J1 = −J2

CollisionsIf IMPUSES are EQUAL, then the MOMEMTUMS of each object are EQUAL as well

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J1 = −J2p1 = −p2mΔv( )1 = −mΔv( )2

m1v11 −m1v1 = −m2v2

1 + m2v2

m1v1 + m2v2 = m1v11 + m2v2

1

The marks are calledprimes and they indicate the velocityafter the collision.

Momentum is CONSERVED

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ΣpBefore = ΣpAfterm1v1 + m2v2 = m1v1

1 + m2v21

In the absence of external forces, the total momentum before a collisionis equal to the total momentum after the collision – The Law of Conservation of Momentum

Example

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Σpbefore = m1v1 + m2v2= 500kg( ) 5 ms( ) + 400kg( ) 2 ms( )

= 2500kg ms + 800kgms = 3300kg

ms

€

Σpafter = m1v11 + m2v2

1

= 500kg( ) 3.0 ms( ) + 400kg( ) 4.5 ms( )

=1500kg ms +1800kgms = 3300kg

ms

Types of Collisions A situation where the objects DO NOT STICK is one

type of collision

Notice that in EACH case, you have TWO objects BEFORE and AFTER the collision.

A “no stick” type collision

Σpbefore

=

Σpafter

1

1

1

22112211

100010000

)10)(3000())(1000(0)20)(1000(

v

v

v

vmvmvmvm oo

-10 m/s

A “stick” type of collision

Σpbefore

=

Σpafter

T

T

T

TToo

v

v

v

vmvmvm

400020000

)4000(0)20)(1000(2211

5 m/s

The “explosion” typeThis type is often referred to as “backwards inelastic”. Notice you have ONE object ( we treat this as a SYSTEM) before the explosion and TWO objects after the explosion.

Backwards Inelastic - Explosions

Suppose we have a 4-kg rifle loaded with a 0.010 kg bullet. When the rifle is fired the bullet exits the barrel with a velocity of 300 m/s. How fast does the gun RECOIL backwards?

Σpbefore

=

Σpafter

2

2

2

2211

430

))(4()300)(010.0()0)(010.4(

v

v

v

vmvmvm TT

-0.75 m/s

Collision SummarySometimes objects stick together or blow apart.

In this case, momentum is ALWAYS conserved.

2211)(

022011

2211022011

vmvmvm

vmvmvm

vmvmvmvm

pp

totalototal

totaltotal

afterbefore

When 2 objects collide and DON’T stick

When 2 objects collide and stick together

When 1 object breaks into 2 objects

Elastic Collision = Kinetic Energy is ConservedInelastic Collision = Kinetic Energy is NOT Conserved

Elastic Collision

JAfterKE

JAfterKE

JmvBeforeKE

car

truck

car

000,50)10)(1000(5.0)(

000,150)10)(3000(5.0)(

000,200)20)(1000(5.021)(

2

2

22

Since KINETIC ENERGY is conserved during the collision we call this an ELASTIC COLLISION.

Inelastic Collision

JAfterKE

JmvBeforeKE

cartruck

car

000,50)5)(4000(5.0)(

000,200)20)(1000(5.021)(

2/

22

Since KINETIC ENERGY was NOT conserved during the collision we call this an INELASTIC COLLISION.

ExampleGranny (m=80 kg) whizzes

around the rink with a velocity of 6 m/s. She suddenly collides with Ambrose (m=40 kg) who is at rest directly in her path. Rather than knock him over, she picks him up and continues in motion without "braking." Determine the velocity of Granny and Ambrose.How many objects do I have before the collision?

How many objects do I have after the collision?

2

1

T

T

TToo

ab

v

v

vmvmvm

pp

120)0)(40()6)(80(2211

4 m/s