A uniform electric field,  $\overrightarrow{E}=-400\sqrt{3}\stackrel{\wedge }{y}N{C}^{-1}$ is applied in a region. A charged particle of mass $m$ carrying positive charge $q$ is projected in this region with an initial speed of  $2\sqrt{10}\times {10}^{6}m{s}^{-1}$. This particle is aimed to hit a target T, which is 5 m away from its entry point into the field as shown schematically in the figure. Take $\frac{q}{m}={10}^{10}Ck{g}^{-1}$. Then

Given : electric field, ${\overrightarrow{E}}_y=-400\sqrt{3}N{C}^{-1}$

Initial speed of charged particle,  $u=2\sqrt{10}\times {10}^{6}m/s$

Range, R = 5 m

F = ma and F = qE  

$\therefore a_y=\frac{q}{m}{E}_y=-400\sqrt{3}\times {10}^{10}$

$\left[\because \frac{q}{m}={10}^{10Ck{g}^{-1}}\left(given\right)\right]$

Range, R$=\frac{u^2\sin 20}{a_y}\Rightarrow 5=\frac{40\times {10}^{12}\sin 20}{400\sqrt{3}\times {10}^{10}}$

$\Rightarrow \sin 20=\frac{\sqrt{3}}{2}$

$\Rightarrow 2\theta ={60}^{o}or,120\Rightarrow \theta ={30}^{o}or,{60}^o$

Particle hits the target, if   $\theta ={30}^{o}or\theta ={60}^o$

                                       $u=2\sqrt{10}\times {10}^{6}m{s}^{-1}$

Time of flight $T_1=\frac{2u\sin \theta }{a_y}=\frac{2\times 2\sqrt{10}\times {10}^6\times \frac{1}{2}}{400\sqrt{3}\times {10}^{10}}$

$=\sqrt{\frac{5}{6}}\mu s\left(for\theta ={30}^o\right)$

Time of flight $T_2=\frac{2u\sin \theta }{a_y}=\frac{2\times 2\sqrt{10}\times {10}^6\times \frac{\sqrt{3}}{2}}{400\sqrt{3}\times {10}^{10}}$

$=\sqrt{\frac{5}{6}}\mu s\left(for\theta ={60}^o\right)$