Answer:
According to the Question:
[tex]\footnotesize\implies F \propto [m]^x [v]^y [r]^z[/tex]
[tex]\footnotesize\implies F =k [m]^x [v]^y [r]^z \:...(i)[/tex]
- K = Dimensionless Constant
Now, substituting Dimensions of each physical quantity in equation (I):
[tex]\footnotesize\implies [M^1L^1T^{-2}] =[M^1L^0T^0]^x [M^0L^1T^{-1}]^y [M^0L^1T^0]^z[/tex]
[tex]\footnotesize\implies [M^1L^1T^{-2}] =[M]^x [L]^{y + z} [T]^{ - y}[/tex]
On comparing the powers of LHS and RHS:
[tex]\implies \bf x = 1[/tex]
[tex]\implies - y = - 2[/tex]
[tex] \implies \bf y = 2[/tex]
[tex]\implies y + z = 1[/tex]
[tex]\implies z = 1 - 2[/tex]
[tex]\implies \bf z = - 1[/tex]
[tex]\footnotesize\implies F = [m]^x [v]^y [r]^z[/tex]
[tex]\footnotesize\implies F = [m]^1 [v]^2 [r]^{ - 1}[/tex]
[tex]\implies\footnotesize \underline{ \boxed{ \bf \red{ F = \frac{m {v}^{2} }{r} }}}[/tex]
