Difference between momentum and kinetic energy

Question

From a mathematical point of view it seems to be clear what's the difference between momentum and $mv$ and kinetic energy $\frac{1}{2} m v^2$. Now my problem is the following: Suppose you want to explain someone without mentioning the formulas what's momentum and what's kinetic energy. How to do that such that it becomes clear what's the difference between those two quantities?

From physics I think one can list essentially the following differences:

1. momentum has a direction, kinetic energy not

2. momentum is conserved, kinetic energy not (but energy is)

3. momentum depends linear on velocity, kinetic energy depends quadratically on velocity

I think is is relatively easy to explain the first two points using everyday language, without referring to formulas.

However is there a good way to illustrate the 3. point?

Answer 1

As a qualitative understanding, here's an example:

If you shoot a bullet, the rifle recoils with the same momentum as the bullet, but the bullet has a lot more Kinetic energy. Aren't you glad your shoulder is being hit by the rifle stock, and not by the bullet?

Answer 2

As a matter of fact, the second point couldn't be more wrong. Momentum is far more ubiquitous than kinetic energy since it is a conserved quantity of every physical system that is translationally invariant.

With respect to your question, user Gerard gave an explanation as intuitive as it gets here