Question

Assume that v=k f l m where k is the constant of proportionality and ABC are number we intend to find by dimension. Prove dimensionally that v=k√fl/m

Answer 1

v=kF^al^bm^c\\ [v]=[k][F]^a[l]^b[m]^c\\ LT^{-1}=(MLT^{-2})^aL^bM^c\\ 1=a+b\\a+c=0\\-1=-2a\to a=\frac{1}{2}\\ b=\frac{1}{2}\\ c=-\frac{1}{2}

$$v=k\sqrt{\frac{Fl}{m}}$$