The Physics
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Opus in profectus

# Conservation of Momentum

## Summary

• Momentum is a conserved quantity.
• The total momentum of a closed system is constant. This principle is known as the law of conservation of momentum (often shortened to the conservation of momentum or momentum conservation).
• When objects interact, their total momentum before the interaction is the same as after the interaction.

pbefore = ∑pafter

• There are several conventions for writing before and after in mathematical shorthand.  ∑p = ∑p′ prime for after ∑p0 = ∑p nought for before ∑q = ∑p q for before ∑pi = ∑pf initial and final
• Numbered subscripts are usually used to identify individual objects.  p1 + p2 + p3 + … = p1′ + p2′ + p3′ + … m1v1 + m2v2 + m3v3 + … = m1v1′ + m2v2′ + m3v3′ + …
• The law of conservation of momentum is logically equivalent to Newton's third law of motion (the action-reaction law).
Related concepts of dynamics
I II
1st law inertia
m
momentum
p = mv
2nd law force law
F = ma
impulse-momentum theorem
J = ∆p
3rd law action-reaction
+F1 = −F2
conservation of momentum
p = ∑p0
• Momentum is a vector quantity, so you need to pay attention to direction.
• Momentums pointing in a convenient direction should be positive.
• Momentums pointing in the opposite direction must be negative.
• Momentums pointing in arbitrary directions are dealt with in another section of this book.
• Recoil
• Recoil occurs when one object moves abruptly backward in reaction to pushing or propelling another object forward.
• The two objects are initially in contact with one another and are therefor at rest relative to one another (p = 0).
• Momentum is conserved, so the total momentum afterwards is still zero (p′ = 0).
• In order for the total momentum to remain zero, the momentum of one object is equal and opposite the other (+p1′ = −p2).
• Collision
• Momentum is conserved for all types of collisions.
• Two objects collide and separate  p1 + p2 = p1′ + p2′ m1v1 + m2v2 = m1v1′ + m2v2′
• Two objects collide and stick together  p1 + p2 = p1+2′ m1v1 + m2v2 = (m1 + m2)v′