Momentum & Energy

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Momentum & Energy Discussion

general info

two objects (1 and 2), velocities before and after (unprime and prime)

conservation of momentum

m1v1 + m2v2 = m1v1 + m2v2

“conservation of kinetic energy” — not a law, just a statement of a possibility

½m1v12 + ½m2v22 = ½m1v12 + ½m2v22

Solve for the velocities after collision. (This is a painful process.) There are two pairs of solutions.

v1 = (m1 − m2)v + 2m2v2
m1 + m2
v2 = (m2 − m1)v + 2m1v1
m1 + m2

or

v1 = v1
v2 = v2

The second pair of solutions says the objects keep going at their original speeds, which implies that they never collided.

Try something. Subtract the other two answers.

v1 − v2 = 
(m1 − m2)v + 2m2v2
m1 + m2

 − 

(m2 − m1)v + 2m1v1
m1 + m2
v1 − v2 = m1v1 − m2v1 + 2m2v2 − m2v2 + m1v2 − 2m1v1
m1 + m2
v1 − v2 = − m2v1 + m2v2 + m1v2 − m1v1
m1 + m2
v1 − v2 = v2(m1 + m2) − v1(m1 + m2)
m1 + m2
v1 − v2 = v2 − v1 
 
1 = v1 − v2
v2 − v1

That’s interesting.

collisions

types of collisions.

c.o.r.typetotal
kinetic energy
comments
0perfectly inelasticdecreases to a minimumobjects stick together
> 0 >inelasticdecreases by any amountall collisions between macroscopic bodies, high energy collisions between subatomic particles
≈ 1 ≈partially elastic, nearly elasticnearly conservedbilliard balls, bowling balls, steel bearings and other objects made from resilient materials
1elasticabsolutely conservedlow energy collisions between atoms, molecules, subatomic particles
> 1 >superelasticincreasescontrived collisions between objects that release potential energy on contact, fictional superelastic materials like flubber
Energy in Collisions

restitution

coefficient of restitution

COR = v1 − v2
v2 − v1

if one of the objects doesn’t move (bouncing a ball of the floor, example) then…

COR = − v
v0

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