Why does potassium react the most vigorously with water?
I interpret "reacts most vigorously" to mean which metal reacts the fastest.  This is a question of kinetics, not thermodynamics (e.g. the answer is not based on the enthalpy of the reaction).  Kinetic rates are determined by the height of the energy barrier that needs to be surmounted, we need to determine and compare the activation energies in this process for the various metals.  The activation energy in this process is determined by the following two steps: $$\ce{M(s) -&gt; M(g)}$$ $$\ce{M(g) -&gt; M^+(g) + e^-}$$ $$\ce{or}$$ $$\ce{M(g) -&gt; M^{+2}(g) + 2 e^-}$$ depending on whether we are analyzing a group I or group II metal.  The following link provides a nice analysis and tabulation of the energies required for these two steps with the group I metals

Group I link

and this link does the same for the Group II metals

Group II link

I'll summarize their findings

Metal ...   Activation Energy (kJ/mol)

K   ...         +508

Na  ...        +603

Mg  ...       +2200

Ca  ...       +1950

CONCLUSION: Potassium has the lowest activation energy and should react the fastest.