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(A) Let the velocity of block $m_1$ be $v$. In case $(a)$,the string is directly connected to $m_2$. Thus,the speed of $m_2$ is $v_1 = v$. In case $(b

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$2 : 1 : 4$

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(A) Let the velocity of block $m_1$ be $v$. In case $(a)$,the string is directly connected to $m_2$. Thus,the speed of $m_2$ is $v_1 = v$. In case $(b)$,the string passes through a movable pulley. If the string moves by $x$,the movable pulley moves by $x/2$. Thus,the speed of $m_2$ is $v_2 = v/2$. In case $(c)$,the string is arranged such that the velocity of $m_2$ is doubled relative to the string velocity. Thus,the speed of $m_2$ is $v_3 = 2v$. The ratio of the speeds is $v_1 : v_2 : v_3 = v : v/2 : 2v = 1 : 0.5 : 2$. Multiplying by $2$,we get $2 : 1 : 4$.

In each of the three arrangements,the block of mass $m_1$ is being pulled left with constant velocity $v$. There is no friction anywhere. The strings are light and inextensible and pulleys are massless. The ratio of the speed of the block of mass $m_2$ in the three cases respectively is

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