The velocity, acceleration and force in two systems of units are related as under:

$\text{ (i) }{\mathrm{v}}^{\mathrm{\prime }}=\frac{{\mathrm{\alpha }}^2}{\mathrm{\beta }}\mathrm{v} \text{ (ii) }{\mathrm{a}}^{\mathrm{\prime }}=\left(\mathrm{\alpha \beta }\right)\mathrm{a} \left(\mathrm{iii}\right) {\mathrm{F}}^{\mathrm{\prime }}=\left(\frac{1}{\mathrm{\alpha \beta }}\right)\mathrm{F}$

All the primed symbols belong to one system and unprinted ones belong to the other system. $\mathrm{\alpha }$ and $\mathrm{\beta }$ arc dimensionless constants. Which of the following is/are correct?

Velocity = length/time, acceleration = length/(time)²

$\begin{array}{l}\Rightarrow \text{ Length }=\frac{(\text{ velocity }{)}^2}{\text{ acceleration }}\text{ i.e., }{\mathrm{L}}^{\mathrm{\prime }}=\frac{{\mathrm{v}}^{\mathrm{\prime }2}}{{\mathrm{a}}^{\mathrm{\prime }}},\mathrm{L}=\frac{{\mathrm{v}}^2}{\mathrm{a}}\\ \Rightarrow \frac{{\mathrm{L}}^{\mathrm{\prime }}}{\mathrm{L}}={\left(\frac{{\mathrm{v}}^{\mathrm{\prime }}}{\mathrm{v}}\right)}^2\left(\frac{\mathrm{a}}{{\mathrm{a}}^{\mathrm{\prime }}}\right)={\left(\frac{{\mathrm{\alpha }}^2}{\mathrm{\beta }}\right)}^2\frac{1}{\mathrm{\alpha \beta }}={\mathrm{\alpha }}^3/{\mathrm{\beta }}^3\end{array}$

$\mathrm{Now}, {\mathrm{m}}^{\mathrm{\prime }}=\frac{{\mathrm{F}}^{\mathrm{\prime }}}{{\mathrm{a}}^{\mathrm{\prime }}},\mathrm{m}=\frac{\mathrm{F}}{\mathrm{a}}$

$\frac{{\mathrm{m}}^{\mathrm{\prime }}}{\mathrm{m}}=\frac{{\mathrm{F}}^{\mathrm{\prime }}}{\mathrm{F}}\frac{\mathrm{a}}{{\mathrm{a}}^{\mathrm{\prime }}}=\frac{1}{\mathrm{\alpha \beta }}\times \frac{1}{\mathrm{\alpha \beta }}=\frac{1}{{\mathrm{\alpha }}^2{\mathrm{\beta }}^2}$

Time = velocity / acceleration, i.e.,

$\begin{array}{l}{\mathrm{T}}^{\mathrm{\prime }}=\frac{{\mathrm{v}}^{\mathrm{\prime }}}{{\mathrm{a}}^{\mathrm{\prime }}}\text{ and }\mathrm{T}=\frac{\mathrm{v}}{\mathrm{a}}\\ \frac{{\mathrm{T}}^{\mathrm{\prime }}}{\mathrm{T}}=\frac{{\mathrm{v}}^{\mathrm{\prime }}}{\mathrm{v}}\frac{\mathrm{a}}{{\mathrm{a}}^{\mathrm{\prime }}}=\frac{{\mathrm{\alpha }}^2}{\mathrm{\beta }}\frac{1}{\mathrm{\alpha \beta }}=\frac{\mathrm{\alpha }}{{\mathrm{\beta }}^2}\end{array}$

Momentum = mass x velocity

${\mathrm{p}}^{\mathrm{\prime }}={\mathrm{m}}^{\mathrm{\prime }}{\mathrm{v}}^{\mathrm{\prime }},\mathrm{P}=\mathrm{mv},\frac{{\mathrm{p}}^{\mathrm{\prime }}}{\mathrm{p}}=\frac{{\mathrm{m}}^{\mathrm{\prime }}}{\mathrm{m}}\frac{{\mathrm{v}}^{\mathrm{\prime }}}{\mathrm{v}}=\frac{1}{{\mathrm{\alpha }}^2{\mathrm{\beta }}^2}\frac{{\mathrm{\alpha }}^2}{\mathrm{\beta }}\frac{{\mathrm{\alpha }}^2}{\mathrm{\beta }}=\frac{1}{{\mathrm{\beta }}^3}$