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 solutions

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

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

Try something. Subtract the two answers.

v1 − v2 =  (m1 − m2)v1  + 2m2v2  −  (m2 − m1)v2  + 2m1v1
m1 + m2 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. type total
kinetic energy
comments
0 perfectly inelastic decreases to a minimum objects stick together
> 0 > inelastic decreases by any amount all collisions between macroscopic bodies, high energy collisions between subatomic particles
≈ 1 ≈ partially elastic, nearly elastic nearly conserved billiard balls, bowling balls, steel bearings and other objects made from resilient materials
1 elastic absolutely conserved low energy collisions between atoms, molecules, subatomic particles
> 1 > superelastic increases contrived 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