Conservation of momentum momentum and impulse

• 
  Define momentum as a vector quantity equal to mass x velocity.
  

• 
  Explain qualitatively, that momentum is conserved in an isolated system
  

There is no truly isolated system on Earth, however, by evaluating a collision of objects right before and right after the collision, and minimizing the amount of friction with outside surfaces, we can show that the quantity ‘mv’ is conserved. This quantity is called momentum. Momentum appears to use the same variables as kinetic energy! However, don’t let this deceive you! Momentum is a vector quantity and therefore the ‘v’ is truly a velocity whereas in kinetic energy it represents speed – energies are scalar!

p = m v where p = momentum (vector quantity – kgm/s),

m= mass (kg), v= velocity (vector quantity - m/s)

eg. Find the momentum of a 125 kg football player running at 4.8 m/s 20^o E of S.

When an object collides with another, it experiences a force of impact during the time of impact. Impulse is the CHANGE in an object’s momentum and is calculated using this force of impact. Impulse for the two objects involved in a collision IS the same in magnitude, but opposite in direction:

p = mv = Ft

• 
  Explain quantitatively, the concepts of impulse and change in momentum using Newton’s laws of motion.
  

eg. A car of mass 2000kg goes from 0-60km/h in 3.5 s. What is the car’s impulse?

Eg. A car (mass 3000kg) hits a brick wall at 45km/h and comes to a complete stop in 0.87 s. What is the force of impact on the car?

Eg. A force of 20.0 N acts north on a 8.00 kg object for 2.5 x 10^-1s. Find the change in velocity of the object.

b. If the object was ORIGINALLY going south at 3.5 m/s, what is its new velocity right after impact?

Force/Time graphs:

A graph depicting force on the ‘y’ axis and time in the ‘x’ axis gives you what result when you find the area???

• 
  Analyze graphs of force and time during a collision
  

When an ‘isolated’collision occurs (as previously defined – ‘isolated system’), the total momentum before collision is equal to the total momentum after collision. This is the LAW OF CONSERVATION OF MOMENTUM and is applied mathematically to collision analysis.

P_before = P_after TOTAL!

• 
  Explain quantitatively that momentum is conserved in one and two-dimensional interactions in an isolated system.
  
• 
  Analyze quantitatively one and two-dimensional interactions
  

Eg. Car A is headed north and collides with Car B that was headed south. The two cars come to a dead stop.

OR: Car A and B have the same headings as in the previous example but after collision cars A and B stick together and head south.

Categories of Collisions:

*momentum is conserved in any kind of collision (that occurs in an isolated system- minimum friction with outside surfaces)

• 
  Define, compare and contrast elastic and inelastic collisions using quantitative examples in terms of conservation of kinetic energy.
  

What has to occur for kinetic energy to be conserved? Physics 20 talked about mechanical energy being conserved but this is more specific to kinetic energy.

–Objects must elastically bounce off of one another. No form of work/potential energy can be created such as dents, abrasions, or breaks otherwise this would not obey the law of conservation of energy (creating more energy than what you started with)

P_before = P_after E_kbefore = E_{k after}

2. Inelastic Collision: Momentum is conserved but kinetic energy is not conserved in the system.

This is much more typical when you think about common collisions. Would there generally be a gain or loss of kinetic energy? Where does that energy go?

-Work is often done on objects, which shows in the form of dents, abrasions or broken pieces. At minimum, a change is the temperature of an object can indicate that work is done. The temperature change shows that there is friction between the colliding objects or within one object as particles collide -this should result in a loss of kinetic energy –loss of energy from the system (conservation of energy).

P_before = P_after E_{k before}≠ E_{k after}

3. Explosions: a special case of inelastic collisions. There is NO kinetic energy before collisions and plenty of it after! Therfore E_k is definitely NOT the same (not conserved). However, momentum is conserved.???

Momentum before is zero! How can it be zero after?

P_before = P_after E_{k before}≠ E_{k after}

TWO-DIMENSIONAL Collisions- When objects that are involved in the collision vary from a linear path EITHER before or after the collision. Independently, they may be traveling in a straight path, but collectively there are angles between the paths.

Eg. Car A is traveling south and Car B is traveling east both towards an uncontrolled intersection. Neither car stops and they collide. The two cars stick/move together.

What approximate direction do you think they will be traveling after collision?

To solve these types of problems we STILL use the law of conservation of momentum, but we do it in a 2D way – vector diagrams and trigonometry! (components)

Eg. A 6.0 kg object traveling north at 5.0 m/s strikes a 4.0 kg object traveling N 30^o W at 7.0 m/s. After the collision the objects remain intact. Find their velocity.