How does energy compares to Momentum? (Besides the direction and the linear & quadratic relationship) This might be a very stupid question; We just learned about Momentum, I got really confused as to the creation and purpose of this unit. I know it "describes motion" with direction, but how so? I know the equations and relation to v (one linear, one quadratic...etc), but why invent this unit in the first place? This unit seems so abrupt and unrelated to all the other ones that were introduced before (Work, Force, etc.)
 A: The question is not stupid at all.
Momentum is involved in Newton's 2nd law,
$$
\boldsymbol{F}=\frac{d}{dt}\left(m\boldsymbol{v}\right)
$$
But more intuitively, it is a measure of the push that an object will transfer you if it hits you. Even if it hits you very slowly, a heavy object can push you very significantly due to its mass. So it makes sense that the product of (mass)x(velocity) has physical meaning.
There are deeper connections farther along the road. You may learn some day that conservation of momentum is connected to the fact that space is the same everywhere.
A: Usually q quantity in physics that one bothered to name, it's because that quantity is very important.
Force is defined in terms of momentum, $\vec F =\dfrac{ \mathrm{d} \vec p}{\mathrm d t}$. This leads to the conservation of momentum, which states that if the net external force on a system is zero, the total momentum of that system will be conserved. The same cannot be said about kinetic energy (only in some cases KE is conserved).
A real life example of the importance of momentum is for instance a car crash. Suppose a car and and plastic bottle are travelling towards you at the same speed. Which one would hurt more? Obviously the car. Why? It has greater momentum. More momentum is transferred from the car that from the plastic bottle, which results in a greater force, which is what you feel.
Kinetic energy and momentum describe different quantities, both extremely important.
In terms of their difference, just think about them as different things that relate to motion. One is a scalar, the other is a vector. In the end, these quantities are both useful for describing many situations.
A: It's a good question, especially since p=mv and T = m$v^2$/2, where T is the kinetic energy, we can write T = $p^2$/2m. So it appears there is a relationship between p and T, and when one is conserved so should the other be conserved so why use both? But this relationship only deals with the magnitude of p. As has been pointed out, p is a vector, p, so has magnitude and direction whereas T is a scalar and only has magnitude. There are situations where p is conserved while T is not, e.g. inelastic collisions, and situations where T is conserved and p is not, e.g. circular motion with constant speed.
