Chapter 8Conservation of Linear Momentum

Linear momentum;Momentum conservation

ImpulseTotal kinetic energy of a system

March 9, 2010

Conservation of Linear Momentum• Definition of linear momentum,

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rp

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rp = mr v Linear momentum is a vector (decompose to x,y,z directions).Units of linear momentum are kg-m/s.

Can write Newton’s second law in terms of momentum:

€

dr p dt

=d(mr v )

dt= m

dr v dt

= mr a

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⇒ dr p dt

=r F net

Momentum force, as if Kinetic energy work

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

u ruu= mi

rvi

i∑ =

rpi

i∑

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 when no net “external” force.(even if “internal” forces are non-conservative).

rFext

i∑ =0

Collisions

“before”m1 m2

“after”m1

m2

momentum before collision = momentum after collision

Always -

But only if

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rF external = 0

Explosion - I“before”M

“after”m1

m2v1

v2

Example: m1 = M/3 m2 = 2M/3

After explosion, which block has larger momentum? (left, right, same)

“before”M

“after”m1

m2v1

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

m2v1

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 speed?

Explosion - I

“before”M

“after”m1

m2v1

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 speed?

mv is the same for each block, so smaller mass has larger speed.

Explosion - I

“before”M

“after”m1

m2v1

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 speed?

mv is the same for each block, so smaller mass has larger speed.

Is kinetic energy conserved?

Explosion - I

Explosion - I“before”M

“after”m1

m2v1

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 speed?

mv is the same for each block, so smaller mass has larger speed.

Is kinetic energy conserved? NO! K was 0 before, it is greater after the explosion.

(internal non-conservative force does some work.)

Momentum and Impulse Momentum

rFaveΔt≡I definition of impulserF =m

ra =m

drv

dt=

drp

dt⇒ Δ

rp=

rFΔt

rp ≡m

rv

For single object…. If F = 0, then momentum conserved (p = 0)

rpsys =

rpi

i∑

Internal forces ≡ forces between objects in system

External forces ≡ any other forces

Δrpsys =

rFextΔt

Thus, if rFext =0, then Δ

rpsys =0

i.e. total momentum is conserved!

• For “system” of objects …

Elastic Collision in 1-Dimension

m1v1i + m2v2i =m1v1 f + m2v2 f

12

m1v1i2 +

12

m2v2i2 =

12

m1v1 f2 +

12

m2v2 f2

Linear momentum is conserved

Energy conserved (for elastic collision only)

Initial Final

Elastic CollisionConservation of Momentum

m1v1i + m2v2i =m1v1 f + m2v2 f

m1(v1i −v1 f ) =m2 (v2 f −v2i )

Conservation of Kinetic Energy

12

m1v1i2 +

12

m2v2i2 =

12

m1v1 f2 +

12

m2v2 f2

m1(v1i2 −v1 f

2 ) =m2 (v2 f2 −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.

Is this an elastic collision?

v1i −v2i =−(v1 f −v2 f )For elastic collision only:

Is this an elastic collision?

v1i −v2i =−(v1 f −v2 f )Yes, the relative speeds are approximately the samebefore and after collision

For elastic collision only:

What is the speed of the golf ball, in case of an elastic collision Club speed: 50 m/sMass of clubhead: 0.5kgMass of golfball: 0.05kgTwo unknowns: after the impact,

speed of club and speed of golfball

Problem solving strategy: - Momentum conservation- Energy conservation (oruse the derived equationfor relative velocities)

v1 f =m1 −m2

m1 + m2

v1i

v2 f =2m1

m1 + m2

v1i

Result:

Special cases:1) Golf shot: m1>>m2

Club speed almost unchangedBall speed almost 2 x club speed

2) Neutron scatters on heavy nucleus: m1<<m2neutron scatters back with almost same speedspeed of nucleus almost unchanged

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

Kinetic energy of a system of particles:

Where

in terms of the CM velocity and relative velocity to the CM.