I have phrased the question as such, to confirm that convolution of the two functions raises the dimensionality of the convolution product. So, if I do convolution of velocity and time, then the resultant should have units of metre x time. Am I correct?
Or an alternative example can be charge Q stored in a capacitor is capacitance (a constant) times the voltage, V, applied across the plates. So, Q(t) = CV(t)
But if the capacitance becomes time varying, then is it correct to say that,
Q(t) = C(t) * V(t),
where * means convolution operation.