The rate law for an elementary step of a reaction of the form $2A \Rightarrow$ Products is given by $rate=k[A]^2$, while in the form $A \Rightarrow$ products is given by $k=[A]$. Is there some sort of intuition as to why the the reaction with $2A$ is quadratic, instead of double the one with $A$? Essentially, why is the equation $k[A]^2$ instead of $2k[A]$?
Intuitively, a reaction between A and B to give C will go twice as fast when you double the concentration of A (because collisions between A and B will occur twice as frequently). Similarly, doubling the concentration of B will also double the reaction rate. But what about the specific case where A and B are the same substance (A + A -> C)? When you double the concentration of A, the rate is quadrupled; when you triple the concentration of A, the rate goes up by a factor of 9, etc.