Momentum and Energy
Discussion
general info
two objects (1 and 2), velocities before and after (unprime and prime)
conservation of momentum
m1v1 + m2v2 = m1v′1 + m2v′2
"conservation of kinetic energy" — not a law, just a statement of a possibility
½m1v12 + ½m2v22 = ½m1v′12 + ½m2v′22
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
v′1 = v1
v′2 = 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.
| v′1 − v′2 = |
|
| v′1 − v′2 = | m1v1 − m2v1 + 2m2v2 |
| 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.
| 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 |
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 |