A block $A$ of mass $M = 2 k g$ is connected to another block $B$ of mass $1 k g$ with a string and a spring of force constant $k = 600 N / m$ as shown in the figure. Initially spring is compressed to $10 c m$ and whole system is moving on a smooth surface with a velocity $v = 1 m / s$ . At any time thread is burnt, the velocity of block $A$ , when $B$ is having maximum velocity w.r.t. ground, is:

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$0$

$1 m / s$

$3 m / s$

$1.5 m / s$

answer is A.

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Detailed Solution

$2 v_{A} + 1 v_{B} = 3 \times 1 \Rightarrow v_{B} = 3 - 2 v_{B}$

$v_{B} \max \Rightarrow \frac{{dv}_{B}}{dt} = 0 \Rightarrow kx = 0$

$\Rightarrow \frac{1}{2} \times 600 \times {(0.1)}^{2} + \frac{1}{2} \times 3 \times 1^{2}$

$= \frac{1}{2} \times 2 v_{A}^{2} + \frac{1}{2} \times 1 v_{B}^{2}$

$\Rightarrow 9 = 2 v_{A}^{2} + v_{B}^{2}$

$\Rightarrow 9 = 2 v_{A}^{2} + {(3 - 2 v_{A})}^{2}$

$\Rightarrow 0 = 6 v_{A}^{2} - 12 v_{A}$

$v_{A} = 2, 0$

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