$v=\lambda^a g^b \rho^c$

using dimension formula

$\Rightarrow\left[ M ^0 L ^1 T ^{-1}\right]=\left[ L ^1\right]^{ a }\left[ L ^1 T ^{-2}\right]^{ b }\left[ M ^1 L ^{-3}\right]^{ c }$

$\Rightarrow\left[ M ^0 L ^1 T ^{-1}\right]=\left[ M ^{ c } L ^{ a + b -3 c } T ^{-2 b}\right]$

$\therefore c =0, a + b -3 c =1,-2 b =-1 \Rightarrow b =\frac{1}{2}$

Now $a+b-3 c=1$

$\Rightarrow a+\frac{1}{2}-0=1$

$\Rightarrow a=\frac{1}{2}$

$\therefore a=\frac{1}{2}, b=\frac{1}{2}, c=0$