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

(A) $\frac{\mathrm{n}^{3}}{\mathrm{~m}^{3}} \mathrm{~L}_{1}=\mathrm{L}_{2}$ and $\frac{\mathrm{n}^{2}}{\mathrm{~m}} \mathrm{~T}_{1}=\mathrm{T}_{2}$

(B) $\mathrm{L}_{1}=\frac{\mathrm{n}^{4}}{\mathrm{~m}^{2}} \mathrm{~L}_{2}$ and $\mathrm{T}_{1}=\frac{\mathrm{n}^{2}}{\mathrm{~m}} \mathrm{~T}_{2}$

(C) $\mathrm{L}_{1}=\frac{\mathrm{n}^{2}}{\mathrm{~m}} \mathrm{~L}_{2}$ and $\mathrm{T}_{1}=\frac{\mathrm{n}^{4}}{\mathrm{~m}^{2}} \mathrm{~T}_{2}$

(D) $\frac{\mathrm{n}^{2}}{\mathrm{~m}} \mathrm{~L}_{1}=\mathrm{L}_{2}$ and $\frac{\mathrm{n}^{4}}{\mathrm{~m}^{2}} \mathrm{~T}_{1}=\mathrm{T}_{2}$

JEE Main · 2022 · 28 Jun 2022 · Shift 2 · Units & Measurement

Velocity (v) and acceleration (a) in two systems of units 1 and 2 are related as

v_{2} = \frac{n}{m ^{2}} v_{1}

and

a_{2} = \frac{a _{1}}{mn}

respectively. Here m and n are constants. The relations for distance and time in two systems respectively are:

(A) $\frac{n ^{3}}{m ^{3}} L_{1} = L_{2}$ and $\frac{n ^{2}}{m} T_{1} = T_{2}$ ✓ Correct
(B) $L_{1} = \frac{n ^{4}}{m ^{2}} L_{2}$ and $T_{1} = \frac{n ^{2}}{m} T_{2}$
(C) $L_{1} = \frac{n ^{2}}{m} L_{2}$ and $T_{1} = \frac{n ^{4}}{m ^{2}} T_{2}$
(D) $\frac{n ^{2}}{m} L_{1} = L_{2}$ and $\frac{n ^{4}}{m ^{2}} T_{1} = T_{2}$

Answer: (A)

From the velocity relation:

\frac{L _{2}}{T _{2}} = \frac{n}{m ^{2}} \frac{L _{1}}{T _{1}}

. From the acceleration relation:

\frac{L _{2}}{T _{2}^{2}} = \frac{L _{1}}{T _{1}^{2} \cdot mn}

. Dividing these two equations gives

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

. Substituting back into the velocity relation yields

\frac{L _{2}}{L _{1}} = \frac{n ^{3}}{m ^{3}}

. Therefore,

\frac{n ^{3}}{m ^{3}} L_{1} = L_{2}

and

\frac{n ^{2}}{m} T_{1} = T_{2}

JEE Main · 2022 · 28 Jun 2022 · Shift 2 · Single correct option