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Kinematics

JEE Main level, single correct, medium

27 of 308

Question 27 of 308

Velocity $\left(v\right)$ and acceleration $\left(a\right)$ in two systems of units $1$ and $2$ are related as ${v}_{2}=\dfrac{n}{{m}^{2}}{v}_{1}$ and ${a}_{2}=\dfrac{{a}_{1}}{mn}$ respectively. Here $m$ and $n$ are constants. The relations for distance and time in two systems respectively are

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Answer: (A) $\dfrac{{n}^{3}}{{m}^{3}}{L}_{1}={L}_{2}$ and $\dfrac{{n}^{2}}{m}{T}_{1}={T}_{2}$

Distance is stated as

[L]=\dfrac{\left[{v}^{2}\right]}{[a]}

\Rightarrow \dfrac{{\left[{v}_{2}\right]}^{2}}{\left[{a}_{2}\right]}=\dfrac{\left[{\left.\dfrac{n}{{m}^{2}}{v}_{1}\right|}^{2}\right.}{{\left[\dfrac{{a}_{1}}{mn}\right]}^{2}}

=\dfrac{{n}^{3}}{{m}^{3}}\dfrac{{\left[{v}_{1}\right]}^{2}}{\left[{a}_{1}\right]}

Or, we have

\left[{L}_{2}\right]=\dfrac{{n}^{3}}{{m}^{3}}\left[{L}_{1}\right]

Similarly, time

\left[T\right]=\dfrac{\left[v\right]}{\left[a\right]}

\Rightarrow \dfrac{\left[{v}_{2}\right]}{\left[{a}_{2}\right]}=\dfrac{\left[\dfrac{n}{{m}^{2}}{v}_{1}\right]}{\left[\dfrac{{a}_{1}}{mn}\right]}=\dfrac{{n}^{2}}{m}\dfrac{\left[{v}_{1}\right]}{\left[{a}_{1}\right]}

So,

{T}_{2}=\dfrac{{n}^{2}}{m}{T}_{1}

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