What's the point of momentum? I know this may sound like a strange question but I have always wondered what exactly the mathematical point of momentum is. I have understood energy to be an attempt to reformulate Newton's 2nd law to be time-independent and to study a system without having to consider time, however, I struggle to find a "goal" for momentum. Forgive me if this question is a little strange.
 A: It doesn't really make sense to talk about a "goal" of any parameter or property in physics. You could invent any name for any group of parameters. Some then turn out to be useful, others not.
So, someone just "happened" to give the group of parameters $mv$ the name momentum and symbol $p$. If that had turned out to be completely useless you would never have heard about it again.
But as it turns out, the group $mv$ is very useful. Its change over time during any impact/interacting turns out to be what we call force, $F=\frac{dp}{dt}$. Momentum change tells us the change in motion taking into account the inertia of the object as well; if you throw an apple upwards and catch it again then in that moment it feels extra "heavy", as if its apparent weight increased; this is momentum. And it furthermore, directly from Newton's 3rd law, turns out that the "amount of" momentum is always constant. That gives us the momentum conservation law, which—just like the energy conservation law—turns out to be very easy to use in mathematical analysis. As a mathematical tool to predict what happens before/during/after collisions, this tool is great - so why not use it?
All in all, any physical concept, term or made-up parameter only has a "goal" if it turns out to be useful for us in some way – if it turns out to be helpful in figuring something out, which of course depends on what we want to figure out. 
A: The point of momentum is that it is conserved.
Any conserved quantity is very important. Practically, conserved quantities often make solving physics problems easy. More importantly, a conserved quantity teaches us something fundamental about how the universe works. The conservation of momentum teaches us that the laws of nature don’t depend on where you are!
This is not obvious, and not covered in beginning physics courses. If you continue in physics you will learn about Noether’s Theorem, which explains how the symmetries of Nature lead to conserved quantities. It is one of the most beautiful things any physicist or mathematician  (Emmy Noether was primarily the latter) has ever discovered.
There are only a handful of known conserved quantities. They are like the rarest of rare jewels. If you were lucky enough to discover a new one, you would be famous forever (well, at least to physicists).
A: In physics generally things aren't defined because one suddenly thought "Hey Lets create something called 'momentum' and lets say that its mass times velocity!"
Of course you can do it.. But the real question is... is the physical quantity that you just defined of any use?
Turns out that's the case with "Momentum".
The general tools for solving various problems will be to exploit the conservation of Momentum and Conservation of energy! But later as you go ahead you slowly start to drop the concept of "conservation of energy" (( Of course its still true )). But you do that because it turns out that Momentum is a more "Fundamental" quantity than energy (This will probably make more sense once you are familiar with the views of quantum mechanics).
