Hi, actually I'm confused about the velocity formula (In blue boundary) why the velocity of that small element taken in that way.
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1 Answer
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Suppose that you have a spring of length $\ell$ fixed at one end and it is extended an amount $e_\ell$ at the other end.
Half way down the spring from the fixed end, $\ell/2$, the extension of the spring is $e_\ell/2$.
In fact the extension of the spring from the fixed end is proportional to the distance from the fixed end.
Thus at a distance $s$ from the fixed end the extension is $e= \dfrac{e_\ell}{\ell} \cdot s$
If one differentiates this expression with respect to time one obtains an expression for the speed of the spring,$\dot e$, at various positions along the spring.
So the speed of the spring a distance $s$ from the fixed end is $\dot e = \dfrac{\dot e_\ell}{\ell} \cdot s = \dfrac{v}{\ell} \cdot s$ where $v\,(=\dot e_\ell)$ is the speed of the spring at the moving end.
