The figure shows one end of a string being pulled down at constant velocity $v$ . Find the velocity of mass ' $m$ ' as a function of ' $x$ '?

- \frac{v}{2 x} \sqrt{x^{2} + b^{2}}

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- \frac{v}{2 x} \sqrt{x^{2} - b^{2}}

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\frac{v}{2 x} \sqrt{x^{2} + b^{2}}

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\frac{v}{2 x} \sqrt{x^{2} - b^{2}}

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Solution

The correct option is A $- \frac{v}{2 x} \sqrt{x^{2} + b^{2}}$
Using constraint equation
$2 \sqrt{x^{2} + b^{2}} + y$ = length of string = constant.
$\frac{2}{2 \sqrt{x^{2} + b^{2}}} \cdot 2 x (\frac{d x}{d t}) + (\frac{d y}{d t}) = 0$
$\Rightarrow (\frac{d y}{d t}) = v (\frac{d x}{d t}) = - \frac{v}{2 x} \sqrt{x^{2} + b^{2}}$

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