Collisions and Conservation of Momentum
Momentum is conserved in a system. See how the conservation of momentum equation is applied to elastic and inelastic collisions.
Name  Variable  MKS Unit  Unit Abbreviation 
Momentum  p  kilograms times meters per second  kg∙m/s 
Mass  m  kilogram  kg 
Velocity  v  meters per second  m/s 
Conservation of Momentum
The more massive an object or faster object has more momentum as seen in the equation.
p = mv
The momentum of objects interacting get conserved. Many scenarios are covered in this section and many examples will represent interaction on a frictionless surface.
Throwing a Projectile
When a player throws a baseball, both the player and baseball start from rest. A baseball player throwing a baseball gains an equal and opposite momentum to the baseball thrown. In reality, friction keeps the player in one place after the throw. In space, a player would accelerate backwards every throw. The amount of acceleration is not as much since the baseball has little mass compared to the player.
Conservation of Momentum Equation
The conservation of momentum equation is not one set in stone. The momentum (p) of all the objects in a system before is equal to the momentum of those objects after. The scenario determines how you break out the conservation of momentum equation.
Example Conservation Scenarios and the Conservation Equation Breakout
Example Conservation of Momentum Scenario’s  Sum of All Momentum Before = Sum of All Momentum After 
Both objects have separate velocities before and after
Two objects colliding and being separate afterwards 
(m_{1}v_{1i}) + (m_{2}v_{2i}) = (m_{1}v_{1f}) + (m_{2}v_{2f})

Both objects are together with the same velocity before but separate after
A person throwing an object, a firecracker blowing up 
(v_{i})(m_{1}+m_{2}) = (m_{1}v_{1f}) + (m_{2}v_{2f})

Both objects are separate before and together traveling one velocity after
A dart hitting a movable target and sticking in it traveling together after 
(m_{1}v_{1i}) + (m_{2}v_{2i}) = (v_{f})(m_{1}+m_{2})

All objects are apart before then the first remains separate from the other two that stick together traveling the same velocity
One car hitting a second that sticks to a third after the collision 
(m_{1}v_{1i}) + (m_{2}v_{2i}) + (m_{3}v_{3i}) = (m_{1}v_{1f}) + (v_{f2+3})(m_{2}+m_{3})

Combined Final Momentum of Two Masses
Notice when objects are together the combined momentum is a single velocity times their masses added together.
(v_{i})(m_{1}+m_{2}) or (v_{f})(m_{1}+m_{2})
Example Problems
1. A 0.80 kg firecracker is traveling through the air at 12 m/s to the right when it explodes. After the explosion, a 0.30 kg piece of it is flying to the left at 6.0 m/s. What is the mass of the other piece and how fast is it flying?
2. A 95 kg pitcher at rest throws a 0.15 kg baseball 40 m/s to the right. How fast would the pitcher be going after the throw on a frictionless surface?
3. How fast is an 85 kg receiver traveling 6 m/s to the right going after catching a 0.43 kg football traveling at 30 m/s right?
Elastic Collisions
In elastic collisions, objects colliding's shape remain unchanged and do not stick together afterwards. Conservation of momentum is conserved and kinetic energy is conserved and no heat given off. Two pool balls colliding on a pool table is an example.
Common Elastic Collision Formula With Two Objects
(m_{1}v_{1i}) + (m_{2}v_{2i}) = (m_{1}v_{1f}) + (m_{2}v_{2f})
Example Problem
4. A 0.1 kg pool ball traveling 2.5 m/s hits another 0.1 kg at rest. If the first ball stops after the elastic collision, how fast is the second now moving?
Inelastic Collisions
During inelastic collisions, objects collide changing form, squishing, and can travel together afterwards. Inelastic collisions involve conservation of momentum but not kinetic energy. Some of the kinetic energy converts to heat as objects change form on impact. You can determine how much kinetic energy has changed by adding up the sum of the kinetic energies before and after (KE = ½ mv^{2})
Common Inelastic Collision Formula With Two Objects Sticking Together
(m_{1}v_{1i}) + (m_{2}v_{2i}) = (v_{f})(m_{1}+m_{2})
Example Problem
5. A 0.05 kg dart traveling 16 m/s hits a 0.15 kg movable target and sticks to it. How fast is the dart in the target moving together after the collision?
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