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→Inelastic Collisions
: '''Conservation of momentum''' means that if you add the [[momentum]] of every [[object]] before and event this will be the same as the total [[momentum]] after that event.
'''Conservation of momentum''' can be applied to 3 types of interaction to allow us to predict the outcome:
*Explosions [[Explosion (Momentum)|Explosion]]s - When two [[object]]s begin with zero [[momentum]] but are [[force]]d apart in opposite directions.*[[Elastic Collisions Collision]]s - When two [[object]]s bounce off one another and the total [[Kinetic Energy Store|kinetic energy]] before a [[Collide|collision ]] is the same as the total [[Kinetic Energy Store|kinetic energy]] after the collision.*[[Inelastic Collisions Collision]]s - When [[Kinetic Energy Store|kinetic energy]] is lost during a [[Collide|collision]], often with the [[object]]s sticking together to form a larger [[object]].
===Equation===
<math>p_{before} = p_{after}</math>
Where:
<math>p_{before}</math> = The total [[momentum]] before an interaction.
<math>p_{before} = p_{after}</math>
Since:
<math>p = mv</math>
<math>0 = m_1 v_1 + m_2 v_2</math>
Where:
<math>m_1</math> = The [[mass]] of [[object]] 1.
<math>p_{before} = p_{after}</math>
Since:
<math>p = mv</math>
<math>m_1 v_1 + m_2 v_2 = m_1 v_3 + m_2 v_4</math>
Where:
<math>m_1</math> = The [[mass]] of [[object]] 1.
<math>v_1</math> = The [[velocity]] of [[object]] 1 before the explosioncollision.
<math>v_3</math> = The [[velocity]] of [[object]] 1 after the explosioncollision.
<math>m_2</math> = The [[mass]] of [[object]] 2.
<math>v_2</math> = The [[velocity]] of [[object]] 2 before the explosioncollision.
<math>v_4</math> = The [[velocity]] of [[object]] 2 after the explosioncollision.
{| class="wikitable"
===Inelastic Collisions===
: During inelastic collisions [[Kinetic Energy Store|kinetic energy]] is not conserved, so the total [[Kinetic Energy Store|kinetic energy]] before is greater than the total [[Kinetic Energy Store|kinetic energy]] after the collision.
: During perfectly inelastic collisions the [[object]]s stick together.
<math>p_{before} = p_{after}</math>
Since
<math>p = mv</math>
Then
{| class="wikitable"
|-
|[[File:ConservationofMomentumInelasticCollision.png|center|600px]]
|}
<math>m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_3</math>
Where
<math>m_1</math> = The [[mass]] of [[object]] 1.
<math>v_1</math> = The [[velocity]] of [[object]] 1 before the collision.
<math>m_2</math> = The [[mass]] of [[object]] 2.
<math>v_2</math> = The [[velocity]] of [[object]] 2 before the collision.
<math>v_3</math> = The [[velocity]] of new larger [[object]] after the collision.
{| class="wikitable"
|-
|[[File:InelasticCollision2.gif|center]]
|}
{| class="wikitable"
| style="height:20px; width: 300px; text-align:center;" |A trolley of [[mass]] 3kg travels at a [[velocity]] of 4m/s before colliding with a trolley of [[mass]] 1kg travelling in the same direction with a [[velocity]] of 2m/s. After the collision the two trolleys stick together. Calculate the [[velocity]] of the new larger trolley after the collision.
| style="height:20px; width: 300px; text-align:center;" |A lorry of [[mass]] 15Mg travels at a [[speed]] of 20m/s towards a car, of [[mass]] 2500kg traveling in the opposite direction at 10m/s. In this collision the two vehicles stick together. Calculate the [[velocity]] of the combined vehicles after the collision correct to two [[Significant Figures|significant figures]].
|-
|[[File:CofMCalc5.png|center|300px]]
|[[File:CofMCalc6.png|center|300px]]
|-
| style="height:20px; width: 300px; text-align:left;" |'''1. State the known quantities'''
m<sub>1</sub> = 3kg
v<sub>1</sub> = 4m/s
m<sub>2</sub> = 1kg
v<sub>2</sub> = 2m/s
| style="height:20px; width: 300px; text-align:left;" |'''1. State the known quantities'''
m<sub>1</sub> = 15Mg = 15x10<sup>3</sup>kg
v<sub>1</sub> = 4m/s
m<sub>2</sub> = 2.5x10<sup>3</sup>kg
v<sub>2</sub> = -10m/s This is negative because it is travelling in the opposite direction.
|-
| style="height:20px; width: 300px; text-align:left;" |'''2. [[Substitute (Maths)|Substitute]] the numbers and [[Evaluate (Maths)|evaluate]].'''
<math>m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_3</math>
<math>3 \times 4 + 1 \times 2 = (3+1) \times v_3</math>
<math>12 + 2 = 4v_3</math>
<math>14 = 4v_3</math>
| style="height:20px; width: 300px; text-align:left;" |'''2. [[Substitute (Maths)|Substitute]] the numbers and [[Evaluate (Maths)|evaluate]].'''
<math>m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_3</math>
<math>15 \times 10^3 \times 20 + 2.5 \times 10^3 \times (-10) = ((15 \times 10^3) + (2.5 \times 10^3)) \times v_3</math>
<math>300000 - 25000 = 17500 \times v_3</math>
<math>275000 = 17500v_3</math>
|-
| style="height:20px; width: 300px; text-align:left;" |'''3. [[Rearrange (Maths)|Rearrange]] the equation and [[Solve (Maths)|solve]].'''
<math>v_3 = \frac{14}{4}</math>
<math>v_3 = 3.5m/s</math>
| style="height:20px; width: 300px; text-align:left;" |'''3. [[Rearrange (Maths)|Rearrange]] the equation and [[Solve (Maths)|solve]].'''
<math>v_3 = \frac{275000}{17500}</math>
<math>v_3 = 15.7142m/s</math>
<math>v_3 \approx 16m/s</math>
|}
===References===
====AQA====
:[https://www.amazon.co.uk/gp/product/1782946403/ref=as_li_tl?ie=UTF8&camp=1634&creative=6738&creativeASIN=1782946403&linkCode=as2&tag=nrjc-21&linkId=32a0abb60dff015b15b50e9b1d7b4644 ''Conservation of momentum, page 183, GCSE Combined Science Trilogy; Physics, CGP, AQA '']
:[https://www.amazon.co.uk/gp/product/1782945970/ref=as_li_tl?ie=UTF8&camp=1634&creative=6738&creativeASIN=1782945970&linkCode=as2&tag=nrjc-21&linkId=a120d24dcc7cc7a58192069a3aafc1d2 ''Conservation of momentum, page 216, GCSE Physics; The Complete 9-1 Course for AQA, CGP, AQA '']
:[https://www.amazon.co.uk/gp/product/019835939X/ref=as_li_tl?ie=UTF8&camp=1634&creative=6738&creativeASIN=019835939X&linkCode=as2&tag=nrjc-21&linkId=57e96876985fc39b1a3d8a3e3dc238b6 ''Conservation of momentum, pages 150-153, 156-157, GCSE Physics; Third Edition, Oxford University Press, AQA '']
:[https://www.amazon.co.uk/gp/product/1471851370/ref=as_li_tl?ie=UTF8&camp=1634&creative=6738&creativeASIN=1471851370&linkCode=as2&tag=nrjc-21&linkId=01c69b0ae058f809cf636033e6ba793e ''Conservation of momentum, pages 169-70m GCSE Physics, Hodder, AQA '']
:[https://www.amazon.co.uk/gp/product/178294558X/ref=as_li_tl?ie=UTF8&camp=1634&creative=6738&creativeASIN=178294558X&linkCode=as2&tag=nrjc-21&linkId=f0dfb66dafcb0c6e9449e7b1a4ae1ac65 ''Conservation of momentum, pages 70, 71, GCSE Physics; The Revision Guide, CGP, AQA '']
====Edexcel====
:[https://www.amazon.co.uk/gp/product/1292120193/ref=as_li_tl?ie=UTF8&camp=1634&creative=6738&creativeASIN=1292120193&linkCode=as2&tag=nrjc-21&linkId=572df39392fb4200db8391d98ae6314e ''Conservation of momentum, law of, page 309, GCSE Combined Science, Pearson Edexcel '']
:[https://www.amazon.co.uk/gp/product/1292120223/ref=as_li_tl?ie=UTF8&camp=1634&creative=6738&creativeASIN=1292120223&linkCode=as2&tag=nrjc-21&linkId=068ecf40278c32406a7f1c6e66751417 ''Conservation of momentum, page 25, GCSE Physics, Pearson Edexcel '']
:[https://www.amazon.co.uk/gp/product/1782945741/ref=as_li_tl?ie=UTF8&camp=1634&creative=6738&creativeASIN=1782945741&linkCode=as2&tag=nrjc-21&linkId=30da4f2178da182547b62a7329d13b57 ''Conservation of momentum, pages 153, 154, GCSE Combined Science; The Revision Guide, CGP, Edexcel '']
:[https://www.amazon.co.uk/gp/product/1782945733/ref=as_li_tl?ie=UTF8&camp=1634&creative=6738&creativeASIN=1782945733&linkCode=as2&tag=nrjc-21&linkId=2a2dbec9db6bf5766c0458d908fa0a52 ''Conservation of momentum, pages 20, 21, GCSE Physics; The Revision Guide, CGP, Edexcel '']
:[https://www.amazon.co.uk/gp/product/1782948163/ref=as_li_tl?ie=UTF8&camp=1634&creative=6738&creativeASIN=1782948163&linkCode=as2&tag=nrjc-21&linkId=0fdbfd5dd397d6e24a9dfb250f08587f ''Conservation of momentum, pages 44, 45, GCSE Physics, CGP, Edexcel '']
====OCR====
:[https://www.amazon.co.uk/gp/product/0198359837/ref=as_li_tl?ie=UTF8&camp=1634&creative=6738&creativeASIN=0198359837&linkCode=as2&tag=nrjc-21&linkId=3c4229e8b023b2b60768e7ea2307cc6f ''Law of conservation; Momentum, pages 72, Gateway GCSE Physics, Oxford, OCR '']
==Key Stage 5==
===Meaning===
The law of '''conservation of momentum''' is the observation that the total [[momentum]] in a [[Closed Isolated System|closed isolated system]] remains the same before and after any interaction between those [[particle]]s (provided no external [[Resultant Force|resultant forces]] acts upon that [[system]].
===About Conservation of Momentum===
*'''Conservation of momentum''' applies to all types of [[Collide|collisions]] and interactions.
*[[Momentum]] is a [[vector]] quantity, meaning it has both [[magnitude]] and direction.
*'''Conservation of momentum''' indicates that the total [[momentum]] before an interaction is equal to the total [[momentum]] after the interaction.
*'''Conservation of momentum''' is a fundamental principle in [[Mechanics (Physics)|mechanics]] and [[Dynamics (Physics)|dynamics]].
*'''Conservation of momentum''' is used to analyse the motion of [[object]]s before and after [[Collide|collisions]].
*'''Conservation of momentum''' is a consequence of [[Newton's Third Law|Newton's third law of motion]].
*In a [[Fundamental Interactions|fundamental interaction]] between [[Subatomic Particle|subatomic particles]] '''momentum is conserved'''. This can be used to detect new [[Subatomic Particle|particles]] which cannot be seen in a cloud chamber. Any time a [[Fundamental Interactions|particle interaction]] appears to violate '''conservation of momentum''' it means that a new, unseen [[Subatomic Particle|particle]] has been created and carried away the [[momentum]].
*'''Conservation of momentum''' is applied to calculations involving [[Elastic Collision|elastic collisions]], [[Inelastic Collision|inelastic collisions]] and [[Explosion (Momentum)|explosions]] to find unknown variables before of after the interaction.
===Formula===
Total Momentum Before = Total Momentum After
<math>p_{before} = p_{after}</math>
Where
<math>p_{before}</math> = The total [[momentum]] before an interaction.
<math>p_{after}</math> = The total [[momentum]] after the interaction.
===Explosions===
: During an explosion a single [[object]] with zero [[momentum]] splits into two smaller [[object]]s.
: The total [[momentum]] before the explosion is zero. Due to '''conservation of momentum''' the total [[momentum]] after the explosion is also zero.
<math>p_{before} = p_{after}</math>
Since
<math>p = mv</math>
Then:
{| class="wikitable"
|-
|[[File:ConservationofMomentumExplosion.png|center|600px]]
|}
<math>0 = m_1 v_1 + m_2 v_2</math>
Where
<math>m_1</math> = The [[mass]] of [[object]] 1.
<math>v_1</math> = The [[velocity]] of [[object]] 1 after the explosion.
<math>m_2</math> = The [[mass]] of [[object]] 2.
<math>v_2</math> = The [[velocity]] of [[object]] 2 after the explosion.
{| class="wikitable"
|-
|[[File:Explosion.gif|center]]
|}
====Example Explosion Calculations====
{| class="wikitable"
| style="height:20px; width:300px; text-align:center;" |An 80kg ice skater and a 90kg ice skater begin at rest and then push away from one another. The 80kg ice skater moves away with a velocity of 0.45m/s. Calculate the [[velocity]] of the 90kg ice skater correct to two [[Significant Figures|significant figures]].
| style="height:20px; width:300px; text-align:center;" |An 18th century cannon of [[mass]] 2000kg fires a 5.5kg cannon ball at a [[velocity]] of 180m/s. Calculate the recoil [[velocity]] of the cannon correct to two [[Significant Figures|significant figures]].
|-
|[[File:CofMCalc1.png|center|300px]]
|[[File:CofMCalc2.png|center|300px]]
|-
| style="height:20px; width:300px; text-align:left;" |'''1. State the known quantities'''
p<sub>before</sub> = 0kgm/s
m<sub>1</sub> = 80kg
m<sub>2</sub> = 90kg
v<sub>1</sub> = 0.45m/s
| style="height:20px; width:300px; text-align:left;" |'''1. State the known quantities'''
p<sub>before</sub> = 0kgm/s
m<sub>1</sub> = 2000kg
m<sub>2</sub> = 5.5kg
v<sub>1</sub> = 180m/s
|-
| style="height:20px; width:300px; text-align:left;" |'''2. [[Substitute (Maths)|Substitute]] the numbers and [[Evaluate (Maths)|evaluate]].'''
<math>p_{before} = p_{after}</math>
<math>0 = m_1 v_1 + m_2 v_2</math>
<math>0 = 80 \times 0.45 + 90 \times v_2</math>
<math>0 = 36 + 90v_2</math>
| style="height:20px; width:300px; text-align:left;" |'''2. [[Substitute (Maths)|Substitute]] the numbers and [[Evaluate (Maths)|evaluate]].'''
<math>p_{before} = p_{after}</math>
<math>0 = m_1 v_1 + m_2 v_2</math>
<math>0 = 2000 \times v_1 + 5.5 \times 180</math>
<math>0 = 2000v_1 + 990</math>
|-
| style="height:20px; width:300px; text-align:left;" |'''3. [[Rearrange (Maths)|Rearrange]] the equation and [[Solve (Maths)|solve]].'''
<math>90v_2 = -36</math>
<math>v_2 = \frac{-36}{90}</math>
<math>v_2 = -0.40m/s</math>
| style="height:20px; width:300px; text-align:left;" |'''3. [[Rearrange (Maths)|Rearrange]] the equation and [[Solve (Maths)|solve]].'''
<math>2000v_1 = -990</math>
<math>v_1 = \frac{-990}{2000}</math>
<math>v_1 = -0.495m/s</math>
<math>v_1 \approx -0.50m/s</math>
|}
===Elastic Collisions===
: During elastic collisions [[Kinetic Energy Store|kinetic energy]] is conserved, so the total [[Kinetic Energy Store|kinetic energy]] before the collision is equal to the total [[Kinetic Energy Store|kinetic energy]] after the collision.
: During elastic collisions the [[object]]s bounce off one another.
<math>p_{before} = p_{after}</math>
Since
<math>p = mv</math>
Then:
{| class="wikitable"
|-
|[[File:ConservationofMomentumElasticCollision.png|center|600px]]
|}
<math>m_1 v_1 + m_2 v_2 = m_1 v_3 + m_2 v_4</math>
Where
<math>m_1</math> = The [[mass]] of [[object]] 1.
<math>v_1</math> = The [[velocity]] of [[object]] 1 before the collision.
<math>v_3</math> = The [[velocity]] of [[object]] 1 after the collision.
<math>m_2</math> = The [[mass]] of [[object]] 2.
<math>v_2</math> = The [[velocity]] of [[object]] 2 before the collision.
<math>v_4</math> = The [[velocity]] of [[object]] 2 after the collision.
{| class="wikitable"
|-
|[[File:ElasticCollision.gif|center]]
|}
====Example Elastic Collision Calculations====
{| class="wikitable"
| style="height:20px; width: 300px; text-align:center;" |A trolley of [[mass]] 3kg travels at a [[velocity]] of 4m/s before colliding with a trolley of [[mass]] 1kg travelling in the same direction with a [[velocity]] of 2m/s. After the collision the 3kg trolley is moving with a [[velocity]] of 3m/s. Calculate the [[velocity]] of the 1kg trolley after the collision.
| style="height:20px; width: 300px; text-align:center;" |A rubber ball of [[mass]] 0.5kg travels at a [[speed]] of 4m/s towards another rubber ball, of [[mass]] 0.2kg traveling in the opposite direction with a [[speed]] of 3m/s. The 0.5kg rubber ball stops completely. Calculate the [[velocity]] of the 0.2kg rubber ball after the collision.
|-
|[[File:CofMCalc3.png|center|300px]]
|[[File:CofMCalc4.png|center|300px]]
|-
| style="height:20px; width: 300px; text-align:left;" |'''1. State the known quantities'''
m<sub>1</sub> = 3kg
v<sub>1</sub> = 4m/s
v<sub>3</sub> = 3m/s
m<sub>2</sub> = 1kg
v<sub>2</sub> = 2m/s
| style="height:20px; width: 300px; text-align:left;" |'''1. State the known quantities'''
m<sub>1</sub> = 0.5kg
v<sub>1</sub> = 4m/s
v<sub>3</sub> = 0m/s
m<sub>2</sub> = 0.2kg
v<sub>2</sub> = -3m/s This is negative because it is travelling in the opposite direction.
|-
| style="height:20px; width: 300px; text-align:left;" |'''2. [[Substitute (Maths)|Substitute]] the numbers and [[Evaluate (Maths)|evaluate]].'''
<math>m_1 v_1 + m_2 v_2 = m_1 v_3 + m_2 v_4</math>
<math>3 \times 4 + 1 \times 2 = 3 \times 3 + 1 \times v_4</math>
<math>12 + 2 = 9 + v_4</math>
<math>14 = 9 + v_4</math>
| style="height:20px; width: 300px; text-align:left;" |'''2. [[Substitute (Maths)|Substitute]] the numbers and [[Evaluate (Maths)|evaluate]].'''
<math>m_1 v_1 + m_2 v_2 = m_1 v_3 + m_2 v_4</math>
<math>0.5 \times 4 + 0.2 \times (-3) = 0.5 \times 0 + 0.2 \times v_4</math>
<math>2 -0.6 = 0 + 0.2v_4</math>
<math>1.4 = 0.2v_4</math>
|-
| style="height:20px; width: 300px; text-align:left;" |'''3. [[Rearrange (Maths)|Rearrange]] the equation and [[Solve (Maths)|solve]].'''
<math>v_4 = 14-9</math>
<math>v_4 = 5m/s</math>
| style="height:20px; width: 300px; text-align:left;" |'''3. [[Rearrange (Maths)|Rearrange]] the equation and [[Solve (Maths)|solve]].'''
<math>v_4 = \frac{1.4}{0.2}</math>
<math>v_4 = 7m/s</math>
|}
===Inelastic Collisions===
: During inelastic collisions [[Kinetic Energy Store|kinetic energy]] is not conserved, so the total [[Kinetic Energy Store|kinetic energy]] before is greater than the total [[Kinetic Energy Store|kinetic energy]] after the collision.
: During perfectly inelastic collisions the [[object]]s stick together.
<math>p_{before} = p_{after}</math>
Since
<math>p = mv</math>
Then
{| class="wikitable"
|-
|[[File:ConservationofMomentumInelasticCollision.png|center|600px]]
|}
<math>m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_3</math>
Where
<math>m_1</math> = The [[mass]] of [[object]] 1.
<math>v_1</math> = The [[velocity]] of [[object]] 1 before the collision.
<math>m_2</math> = The [[mass]] of [[object]] 2.
<math>v_2</math> = The [[velocity]] of [[object]] 2 before the collision.
<math>v_3</math> = The [[velocity]] of new larger [[object]] after the collision.
{| class="wikitable"
|-
|[[File:InelasticCollision2.gif|center]]
|}
{| class="wikitable"
| style="height:20px; width: 300px; text-align:center;" |A trolley of [[mass]] 3kg travels at a [[velocity]] of 4m/s before colliding with a trolley of [[mass]] 1kg travelling in the same direction with a [[velocity]] of 2m/s. After the collision the two trolleys stick together. Calculate the [[velocity]] of the new larger trolley after the collision.
| style="height:20px; width: 300px; text-align:center;" |A lorry of [[mass]] 15Mg travels at a [[speed]] of 20m/s towards a car, of [[mass]] 2500kg traveling in the opposite direction at 10m/s. In this collision the two vehicles stick together. Calculate the [[velocity]] of the combined vehicles after the collision correct to two [[Significant Figures|significant figures]].
|-
|[[File:CofMCalc5.png|center|300px]]
|[[File:CofMCalc6.png|center|300px]]
|-
| style="height:20px; width: 300px; text-align:left;" |'''1. State the known quantities'''
m<sub>1</sub> = 3kg
v<sub>1</sub> = 4m/s
m<sub>2</sub> = 1kg
v<sub>2</sub> = 2m/s
| style="height:20px; width: 300px; text-align:left;" |'''1. State the known quantities'''
m<sub>1</sub> = 15Mg = 15x10<sup>3</sup>kg
v<sub>1</sub> = 4m/s
m<sub>2</sub> = 2.5x10<sup>3</sup>kg
v<sub>2</sub> = -10m/s This is negative because it is travelling in the opposite direction.
|-
| style="height:20px; width: 300px; text-align:left;" |'''2. [[Substitute (Maths)|Substitute]] the numbers and [[Evaluate (Maths)|evaluate]].'''
<math>m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_3</math>
<math>3 \times 4 + 1 \times 2 = (3+1) \times v_3</math>
<math>12 + 2 = 4v_3</math>
<math>14 = 4v_3</math>
| style="height:20px; width: 300px; text-align:left;" |'''2. [[Substitute (Maths)|Substitute]] the numbers and [[Evaluate (Maths)|evaluate]].'''
<math>m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_3</math>
<math>15 \times 10^3 \times 20 + 2.5 \times 10^3 \times (-10) = ((15 \times 10^3) + (2.5 \times 10^3)) \times v_3</math>
<math>300000 - 25000 = 17500 \times v_3</math>
<math>275000 = 17500v_3</math>
|-
| style="height:20px; width: 300px; text-align:left;" |'''3. [[Rearrange (Maths)|Rearrange]] the equation and [[Solve (Maths)|solve]].'''
<math>v_3 = \frac{14}{4}</math>
<math>v_3 = 3.5m/s</math>
| style="height:20px; width: 300px; text-align:left;" |'''3. [[Rearrange (Maths)|Rearrange]] the equation and [[Solve (Maths)|solve]].'''
<math>v_3 = \frac{275000}{17500}</math>
<math>v_3 = 15.7142m/s</math>
<math>v_3 \approx 16m/s</math>
|}