Is the law of conservation of momentum underpinned by the law of conservation of energy?
No . They both follow from Newtons laws at the most basic level. Also note that Energy is a scalar quantity and momentum is a vector, so there is quite a bit of difference in the kind of information they provide.
also, momentum is sometime more easy to keep track of, than energy. So in some situations we use a physical model where energy is not conserved, but momentum is conserved. For example, inelastic collisions, where we say momentum must be conserved, but we do not require energy to be conserved. In reality, the energy is lost as heat and deformation of the objects. But since it is hard to keep track of those things, we often just model the situation as if it does not conserve energy.
Nice example of this:
haha, yeah I saw that. It felt nice to guess the right answer straight away. I suppose it is the intuition that comes from doing these kinds of problems many times. Momentum is more important than energy, when it comes to these types of problems. Again, I feel the need to say energy is 'truly' conserved too. But it's lost as heat or deformation, which is not easy to identify in these kinds of problems.
underpinned by Newton's 3rd law.
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