In the region $0 < x < L$, there is an electric field $\vec{E}=E_{0} \hat{x}$, where $\hat{x}$ denotes the unit vector parallel to the $x$-axis. Two small balls, each of mass $m$ and carrying charge $q$ $\left(q E_{0}>0\right)$ are connected with a weightless non-stretchable string of length $l$. Initially, at the moment of time $t=0$, the string is taut, the velocity of the both balls is $\vec{v}=v_{0} \hat{x}$, one of the balls, the ball $A$, is at $x=0$ while the other ball, the ball $B$ is at $x<0$. The electric field created by the balls can be neglected, and it can be assumed that $v_{0}$ is very small (much smaller than $\sqrt{E L q / m}$ ).

i  ^2.00 Consider the case when the string is parallel to the $x$-axis, and $l=L$ Sketch the dependence of the velocity of the both balls as a function of time. Will the balls collide? If yes then when?

ii  ^2.00 Now, at $t=0$, the string forms an angle of $45^{\circ}$ with the $x$-axis, $l=1.2291 L$. By this string length, at the moment $t=T$ when the ball $A$ reaches $x=L$, the string is parallel to the $x$-axis. Find $T$.

iii  ^2.00 Under the assumptions of the previous task, what is the speed of the ball $A$ at the moment $t=T$?

iv  ^2.00 Now, at $t=0$, the string forms a very small angle $\phi$ with the $x$-axis. Similarly to the previous two tasks, at the moment $t=T^{\prime}$ when the ball $A$ reaches $x=L$, the string is parallel to the $x$-axis. Find the string length $l$ assuming that $l>L$. The answer to this point should contain only $L$ and numerical constants.