Question

In two systems of relations among velocity, acceleration, and force are, respectively, $v_2=\frac{{\alpha }^2}{\beta }{v}_1,a_2=\alpha \beta a_1$ and $F_2=\frac{F_1}{\alpha \beta }$. If $\alpha$ and $\beta$ are constants, then make relations among mass, length, and time in two systems.

Answer 1

Relation between mass :

From the equation, $F_2=\frac{F_1}{\alpha \beta }$

$M_2{a}_2=\frac{M_1{a}_1}{\alpha \beta }$

Or $M_2\times \left(\alpha \beta a_1\right)=\frac{M_1{a}_1}{\alpha \beta }$

$⟹M_2=\frac{M_1}{{\alpha }^2{\beta }^2}$

Relation between Time :

Time $T=\frac{v}{a}$

So, $T_2=\frac{v_2}{a_2}=\frac{{\alpha }^2{v}_1/\beta }{\alpha \beta a_1}$

Or $T_2=\frac{\alpha }{{\beta }^2}{T}_1$ $(\frac{v_1}{a_1}=T_1)$

Relation between Length :

Length $L=vT$

So, $L_2=v_2{T}_2=\frac{{\alpha }^2}{\beta }{v}_1\times \frac{\alpha }{{\beta }^2}{T}_1$

$⟹L_2=\frac{{\alpha }^3}{{\beta }^3}{L}_1$