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<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:InelasticCollision.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>
|}