How can I interpret c^2 or v^2?

Okay, reading through the other answers it seems as if the issue lies with not understanding what energy is, so I will start there.

Every time we model nature using a symmetry, there is ALWAYS a corresponding conserved quantity. This is called Noether's Theroem. What this means is rather difficult to understand at first, so I will give some examples:

*If a physical system exhibits the same outcomes regardless of its location is space, then linear momentum is conserved. *If a system exhibits the same outcomes regardless of its orientation in space, then angular momentum is conserved. *If a system exhibits the same results regardless of time, then energy is the value which is conserved.

What this tells us is that energy is "nothing more" than a mathematical necessity. It is a quantity which MUST exist and stay constant for any system so long as that system has symmetry in time (i.e. if you are standing on top of a tower and you drop a ball, the system is symmetric in time so long as the ball falls the same no matter when you decide to drop it). Not only is energy constant, it is also a scalar! Going back to our ball and tower: if the tower is fixed to the earth, which is rotating, the velocity vector of the falling ball from an earth-centered frame would be exactly opposite 12 hours apart (since your tower is now "pointing" in the opposite direction due to the earth having completed a half revolution). If energy were also a vector, it would not be constant in this situation, but it is because it is a scalar and cares not for direction.

Now what on earth does this have to do with multiplying two velocities together? What you have to remember is that a velocity is a vector. Which means it has magnitude AND direction. Energy on the other hand, is a scalar. It MUST be a scalar, due to Noether's Theorem. This means that in order to go from a velocity to an energy, you need to take the inner product of the velocity. For a one dimensional velocity this translates to a v2 , but for a 3d velocity it is actually [v,v]. What this means is that you are "filtering out" direction. Given a kinetic energy of an object with known mass it is impossible to determine the velocity, simply because while we know the magnitude of the velocity the information of the direction has been lost BECAUSE direction is NOT time symmetric, and in order for energy to be the conserved value of the time symmetric system we must lose the direction component of the velocity.

So how to visualize it? Well, we are creatures of time, so imagining things in a time invariant world for us is hard. Therefore while it is easy to understand energy in the context of a constant force applied over a specific distance, it is hard for us to imagine what a velocity times a velocity, or more accurately a momentum times a velocity, "means". So in an hand wavy sense I guess you could say that kinetic energy "means" the velocity of momentum.

/r/askscience Thread