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4.1a Further Mechanics

Momentum conceptsBreithaupt pages 4 to 17

September 3rd, 2010

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AQA A2 Specification

Lessons Topics

1 to 4 Momentum concepts

Force as the rate of change of momentum

F = (mv) / t

ImpulseFt = (mv)

Significance of area under a force-time graph.

Principle of conservation of linear momentum applied

to problems in one dimension.Elastic and inelastic collisions; explosions.

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Momentum, p

momentum = mass x velocityp = mv

min kilograms (kg)vin metres per second (ms-1)

pin kilograms metres per second (kg ms-1)

Momentum is a VECTOR quantity

direction the same as the velocity

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Newtons 2ndlaw (A2 version )

The resultant force acting on an object is

proportional to the rate of change of

momentum of the object and is in the same

direction as the resultant force.

F (p)

(t)

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Inserting a constant of proportionality k

F= k(p)

(t)

but: p = mv

hence: F= k(mv)

(t)

If the mass, m remains constant:

F= km(v)(t)

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but: (v) = a (acceleration)

(t)

hence: F= km aA force of one newton is defined as that required tocause an acceleration of 1 ms -2 with a mass of 1 kg.

Inserting these values into:F= km a

gives: 1 = kx 1 x 1

and so: k= 1

giving: F = m a (the AS version of Newtons 2nd law)

Note: This simplified version only applies for an objectofconstant mass.

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Force and Momentum

Force is equal to the rate of change of

momentum.

F =(mv) /t

F in newtons (N)(mv) in kilograms metres per second (kg ms-1)

t in seconds (s)

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Question 1

A car of mass 800 kgmoving at a velocity of30 ms-1 is brought torest by a braking forceof 1200 N.

Calculate:(a) its initial momentum

(b) the time taken tostop the car.

(a) p = mv= 800 kg x 30 ms-1momentum = 24 000 kg ms-1

(b) F =(mv) /t1200N = 24 000 kg ms-1 /t

t= 24 000 kg ms-1 / 1200Ntime = 20 seconds

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Question 2

A car of mass 750kg

travelling at a speed of4.0ms-1 is struck frombehind by another vehicle.The impact lasts for 0.30sand causes the speed of thecar to increase to 6.0ms-1 .

Calculate:(a) the change inmomentum of the car due tothe impact.

(b) the impact force.

(a) p =mv

mass is constant, so:

p = mv

= 750 kg x (6.04.0) ms-1

momentum change= 1 500 kg ms-1

(b) F =(mv) /t

= 1 500 kg ms-1 / 0.30sforce = 5000 N

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Impulse,p

Impulse is equal to the change ofmomentum produced by a force over aperiod of time.

Impulse, p = Ft

= (mv)

p is measured in newton seconds (Ns)

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Impulse caused by a golf club(Breithaupt page 8)

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Force time graphs(Breithaupt page 6)

Impulse is

equal to the

area under aforce-time

graph.

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Calculation Example(Breithaupt page 9)

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Graph Question

Calculate the impulse andchange in velocity caused to

mass of 6kg from the graph

opposite.

Area = impulse= 3N x (5 - 2 )s

impulse = 9 Ns

=(mv) = 6kg x(v)

therefore,(v) = 9 / 6

velocity change = 1.5 ms-12 5 t/ s

3.0

F/ N

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Conservation

of Linear Momentum

The total linear momentum of an isolated

system of bodies remains constant

An isolated system is one where no external forces

(e.g. friction or air resistance) acts on the interacting

bodies.

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Question

A trolley of mass 4kg moving at 5ms-1 collides withanother initially stationary trolley of mass 3kg. Ifafter the collision the trolleys move off attachedtogether calculate their common final velocity.

Initial total linear momentum of the system:= momentum of 4kg trolley + momentum of 3kg trolley

= (4kg x 5ms-1) + (3kg x 0ms-1)

= 20 kgms-1

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Conservation of linear momentum:

Final total linear momentum of the system

must also = 20 kgms-1

(total mass x final common velocity) = 20 kgms-1

(4kg + 3kg) x v = 20 kgms-1

7v = 20

v = 20 / 7

Final common velocity = 2.86 ms-1

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Elastic and inelastic collisions

ELASTICKINETIC energy is conserved

INELASTIC Some (or all) KINETIC energyis transformed into thermal or other forms of

energy.

In both types of collision both the total

energy and momentum are conserved.

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Collision question continued

Was the col l is ion in the previous examp le

elast ic o r inelast ic ?

Kinetic energy = x mass x (speed)2

Total initial KE = KE of 4kg trolley

= x 4kg x (5 ms-1)2 = 2 x 25 = 50 JTotal final KE = KE of combined 7kg trolley

= x 7kg x (2.86 ms-1)2 = 3.5 x 8.18 = 28.6 J

Kinetic energy reduced Collision INELASTIC

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Explosions

KINETICenergy isincreased

Both the totalenergy andmomentum areconserved

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Question

A gun of mass 3kg f i res a bul let of mass 15g.If the bul let moves o ff at a speed o f 250ms-1calculate the reco i l speed o f the gun .

Initial total linear momentum of the system:= momentum of the gun + momentum of the bullet

= (3kg x 0ms-1) + (15g x 0ms-1)

= 0 kgms-1

Conservation of linear momentum: Final total

linear momentum of the system must also = 0kgms-1

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Therefore:

(bullet mass x velocity) + (gun mass x velocity) = 0

(0.015kg x 250ms-1) + (3kg x gun velocity) = 0(3.75) + (3 x gun velocity) = 0

3 x gun velocity = - 3.75

gun velocity = - 3.75 / 3 = - 1.25 ms-1

The MINUS sign indicates that the guns velocity isin the opposite direction to that of the bullet

Gun recoil speed = 1.25 ms-1

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Internet Links

Effect of impulse - NTNU

Collisions along a straight line - NTNU

1D collision showing momentum and ke - NTNU

2D collisions - NTNU

2D Collisions - Explore Science

Two dimensional collisions - Virginia

Elastic & Inelastic Collisions - Fendt

Newton's Cradle - Fendt

Gaussian gun - NTNU

Dropping a load onto a trolley - momentum - netfirms

Ballistic Pendulum - NTNU

http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=400.0http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=20http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=20http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=193.0http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=193.0http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=120.0http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=120.0http://www.ionaphysics.org/lab/Explore/dswmedia/2dcollis.htmhttp://www.ionaphysics.org/lab/Explore/dswmedia/2dcollis.htmhttp://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/Collision/jarapplet.htmlhttp://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/Collision/jarapplet.htmlhttp://www.walter-fendt.de/ph11e/collision.htmhttp://www.walter-fendt.de/ph11e/collision.htmhttp://www.walter-fendt.de/ph11e/ncradle.htmhttp://www.walter-fendt.de/ph11e/ncradle.htmhttp://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=203.0http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=203.0http://www.ngsir.netfirms.com/englishhtm/DropABrick.htmhttp://www.ngsir.netfirms.com/englishhtm/DropABrick.htmhttp://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=448.0http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=448.0http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=448.0http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=448.0http://www.ngsir.netfirms.com/englishhtm/DropABrick.htmhttp://www.ngsir.netfirms.com/englishhtm/DropABrick.htmhttp://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=203.0http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=203.0http://www.walter-fendt.de/ph11e/ncradle.htmhttp://www.walter-fendt.de/ph11e/ncradle.htmhttp://www.walter-fendt.de/ph11e/collision.htmhttp://www.walter-fendt.de/ph11e/collision.htmhttp://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/Collision/jarapplet.htmlhttp://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/Collision/jarapplet.htmlhttp://www.ionaphysics.org/lab/Explore/dswmedia/2dcollis.htmhttp://www.ionaphysics.org/lab/Explore/dswmedia/2dcollis.htmhttp://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=120.0http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=120.0http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=193.0http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=193.0http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=20http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=20http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=400.0

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Core Notes from Breithaupt pages 4 to 171. Define what is meant by momentum. State its unit.

2. Explain how force is related to the rate of change of momentum.3. What is meant by impulse? How can impulse be found

graphically? Copy Figure 3 on page 6.

4. State the principle of conservation of momentum.

5. Redo the worked example on page 13 this time with the first railwagon moving at 4ms-1 colliding with another now of mass 2000kg.

6. Define what is meant by (a) an elastic and (b) an inelastic collision.

7. Redo the worked example on page 15 this time with the first railwagon moving at 4ms-1 colliding with another now of mass 6000kg.

8. Explain how the principle of conservation of momentum applies inan explosion.

9. Redo summary question 1 on page 17 this time with a shell of mass3.0kg being fired from a gun of mass 1000kg.

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Notes from Breithaupt pages 4 to 7

Force & Momentum

1. Define what is meant by momentum. State its unit.

2. Explain how force is related to the rate of change ofmomentum.

3. What is meant by impulse? How can impulse be foundgraphically? Copy Figure 3 on page 6.

4. Show how the version of Newtons 2nd law of motion onpage 5 can be used to derive the equation: F = ma

5. Redo the worked example on page 7 this time with a

force of 20N on a mass of 200kg.6. Try the summary questions on page 7

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Notes from Breithaupt pages 8 to 10

Impact Forces

1. Redo the worked example on page 8 this time with avelocity increase of 25ms-1 over a time of 20ms.

2. Explain how the relationship between force andmomentum change is relevant to vehicle safety.

3. Explain why a greater force is needed to send a ballback along its initial path than to deflect it at an angle.

4. Redo the worked example on page 10 this time with aball of mass 0.30kg moving at an initial speed of15ms-1.

5. Try the summary questions on page 10

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Notes from Breithaupt pages 11 to 13

Conservation of momentum

1. State the principle of conservation of momentum.2. Redo the worked example on page 13 this time with the

first rail wagon moving at 4ms-1 colliding with anothernow of mass 2000kg.

3. Show how the version of Newtons 3rd law of motion onpage 11 can be used to derive the principle ofconservation of momentum.

4. Explain how the principle of conservation of momentumcan be verified experimentally.

5. Explain how the principle of conservation of momentumapplies in a head-on collision.

6. Try the summary questions on page 13

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Notes from Breithaupt pages 14 & 15

Elastic and inelastic collisions

1. Define what is meant by (a) an elastic and (b) aninelastic collision.

2. Redo the worked example on page 15 this time with thefirst rail wagon moving at 4ms-1 colliding with another

now of mass 6000kg.

3. What is a totally inelastic collision?

4. Explain how conservation of energy can still apply to an

inelastic collision. What else is conserved in this type ofcollision?

5. Try the summary questions on page 15

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Notes from Breithaupt pages 16 & 17

Explosions

1. Explain how the principle of conservation ofmomentum applies in an explosion.

2. Redo summary question 1 on page 17 this timewith a shell of mass 3.0kg being fired from a

gun of mass 1000kg.

3. Explain how the principle of conservation ofmomentum can be verified experimentally with

an explosive interaction.4. Try the other summary questions on page 17