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(D) Let the initial positions of balls $A$ and $B$ be $(0, 0)$ and $(L, 0)$ respectively. At $t = 0$,ball $A$ is given velocity $v$ along the $x$-axis

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Both $(A)$ and $(C)$

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(D) Let the initial positions of balls $A$ and $B$ be $(0, 0)$ and $(L, 0)$ respectively. At $t = 0$,ball $A$ is given velocity $v$ along the $x$-axis,so its position at time $t$ is $(vt, 0)$. Ball $B$ is given velocity $v$ along the $y$-axis,so its position at time $t$ is $(L, vt)$.The distance $d$ between the balls at time $t$ is given by $d^2 = (L - vt)^2 + (vt - 0)^2 = L^2 - 2Lvt + v^2t^2 + v^2t^2 = L^2 - 2Lvt + 2v^2t^2$.The thread becomes taut when the distance between the balls equals the length of the thread,i.e.,$d = L$.So,$L^2 = L^2 - 2Lvt + 2v^2t^2$.$2v^2t^2 - 2Lvt = 0$.$2vt(vt - L) = 0$.This gives $t = 0$ or $t = L/v$.At $t = L/v$,the distance between the balls again becomes $L$,and for $t > L/v$,the distance $d$ increases beyond $L$,meaning the thread remains taut.

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