Putting the equations together and resolving components...

The Y-direction equation becomes:

Y:

The X-direction equation becomes:

X:

Since we are interested in finding the maximum velocity before skidding we can require that the frictional force have its maximum value, ie F_{frictional = FStatic = mN}

so our two equations become:

Y:
X:

Which can be combined to give:

and which simplifies to: