The Physics
Hypertextbook
Opus in profectus

# Momentum and 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.

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

That's interesting.

### collisions

types of collisions.

Energy in collisions
c.o.r. type total
kinetic energy
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

### restitution

coefficient of restitution

 COR = v′1 − v′2 v2 − v1

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

 COR = − v v0