Sometimes it is convenient to construct a system of units so that all quantities can be expressed in

Question

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity $X$ as follows: [position] $=\left[X^{\alpha}ight]$; [speed] $=\left[X^{\beta}ight]$; [acceleration] $=\left[X^{p}ight]$; [linear momentum $]=\left[X^{q}ight] ;[$ force $]=\left[X^{r}ight]$. Then\

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(a) : Given $[$ position $]=\left[X^\alphaight]$ [speed] $=\left[X^\betaight]$ [acceleration] $=\left[X^pight]$; [linear momentum] $=\left[X^qight]$ [force] $=\left[X^right]$ Now, \ $\frac{	ext { linear momentum }}{	ext { force }}=\frac{m v}{m a}=\frac{v}{a}=\left[X^{q-r}ight]$ But $\frac{	ext { speed }}{	ext { acceleration }}=\frac{v}{a}=\left[X^{\beta-p}ight]$ $	herefore \quad q-r=\beta-p$ or $\beta=q+p-r$ \  Now from equation of motion, $(	ext { speed })^2=2($ acceleration $) 	imes($ position $)$ $X^{2 \beta}=X^{p+\alpha}$ or $\alpha+p=2 \beta$ ```