: '''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===
{| class="wikitable"
|-
|[[File:InelasticCollisionInelasticCollision2.gif|center]]
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==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 all any interaction between those [[particle]]s (provided no external [[Fundamental InteractionsResultant Force|fundamental interactionsresultant forces]] between acts upon that [[particlesystem]]s.
===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>|}