What has to be true to use conservation of momentum vs. energy

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Conservation of momentum applies when no external forces act on a system, requiring a net force of zero in each direction for momentum to be conserved. Common examples include collisions between objects and interactions on frictionless surfaces, such as ice skaters or boats pushing off each other. Conservation of energy occurs when no external work is done on the system, with typical external work sources being friction and air resistance. While momentum is conserved in all types of collisions, kinetic energy is only conserved in elastic collisions; inelastic collisions result in energy loss due to deformation. Understanding these principles is crucial for analyzing physical systems effectively.
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We are currently learning about conservation of momentum and conservation of energy in my intro physics class @ berkeley.

What must be true for me to use one of the other or both (I'd appreciate some common examples of when to use/when not to use them too, if possible)
 
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Conservation of momentum holds for a system when there are no external forces on the system. Since momentum is a vector, we need to check the net force in each direction, and if the net force is zero then the momentum for that direction is conserved.

The primary examples in intro physics are collisions (between blocks, disks, sports equipment, bullets and targets, etc) and also frictionless surface problems like ice skaters who push away from each other or people in boats that push away from each other, etc.

Conservation of energy in a system occurs whenever there is no external work done on the system. Typical causes of external work are friction, air resistance, or normal pushing / pulling.
 
Momentum is conserved in all collisions. Kinetic energy is only conserved in elastic collisions. In the case of inelastic collisions, kinetic energy is lost due to deformation of the objects involved in the collision.
 
Thanks so much for the fast responses! Makes sense!
 
I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...
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