A particle is moving in x-y plane according to the curve $\mathrm{x}=4{\mathrm{y}}^2$ such that $\mathrm{y}=\frac{{\mathrm{t}}^3}{3}$ where x & y are in metres & t is in sec. The speed v of the particle at t=1sec is $\frac{\sqrt{\mathrm{n}}}{3}{\mathrm{ms}}^{-1}$then value of n is

$\text{ (A) }\mathrm{x}=4{\mathrm{y}}^2=4{\left[\frac{{\mathrm{t}}^3}{3}\right]}^2=\frac{4{\mathrm{t}}^6}{9}\text{ and }\mathrm{y}=\frac{{\mathrm{t}}^3}{3}$
velocity of particle at any time t is,
$\begin{array}{l}\overline{\mathrm{v}}=\frac{\mathrm{dx}}{\mathrm{dt}}\widehat{\mathrm{i}}+\frac{\mathrm{dy}}{\mathrm{dt}}\widehat{\mathrm{j}}\\ \Rightarrow \overline{\mathrm{v}}=\frac{8{\mathrm{t}}^5}{3}\widehat{\mathrm{i}}+{\mathrm{t}}^2\widehat{\mathrm{j}}\end{array}$
Velocity of particle at t=1s,$\overline{\mathrm{v}}=\frac{8}{3}\widehat{\mathrm{i}}+\widehat{\mathrm{j}}$
So speed of particle at t =1s is,
$\mathrm{v}=\sqrt{{\left(\frac{8}{3}\right)}^2+1}=\frac{\sqrt{73}}{3}=2.84\mathrm{m}/\mathrm{s}$