Given that  $\frac{{\text{a}}_{\text{t}}}{\text{t}}={\text{tan 60}}^{\text{o}}$   

$\Rightarrow {\text{a}}_{\text{t}}=\sqrt{3}\text{t} \dots \left(1\right)$

$\frac{\text{dv}}{\text{dt}}=\sqrt{3}\text{t}$                                

$\Rightarrow \underset{0}{\overset{\text{v}}{\int }}\text{dv}=\sqrt{3}\text{t dt}$



$\text{v}=\frac{\sqrt{3}{\text{t}}^2}{2}$            ...(2)

Given that   ${\text{tan 30}}^{\text{o}}=\frac{{\text{a}}_{\text{t}}}{{\text{a}}_{\text{c}}} \Rightarrow \frac{1}{\sqrt{3}}=\frac{{\text{a}}_{\text{t}}}{{\text{a}}_{\text{c}}}$

$\Rightarrow {\text{a}}_{\text{c}}=\sqrt{3}{\text{a}}_{\text{t}}$

$\frac{{\text{v}}^2}{\text{R}}=\sqrt{3}{\text{a}}_{\text{t}} \dots \left(3\right)$

From (1) and (2)  $\frac{{\left(\frac{\sqrt{3}}{2}{\text{t}}^2\right)}^2}{1}=\sqrt{3}\times \sqrt{3}\text{t} \Rightarrow \frac{3}{4}{\text{t}}^4=3\text{t}$

${\text{t}}^3=4$                

$\Rightarrow \text{t}=2^{2/3}\text{sec.}$