# Frequency of vibration of a spring Hi, actually I'm confused about the velocity formula (In blue boundary) why the velocity of that small element taken in that way. • Please avoid screenshots and write a self-contained question (a link or reference to the book you based it on is still possible of course). It'll allow you to focus on the exact problem and nothing else. Jul 18, 2022 at 16:25
• We use Mathjax as the site standard for displaying math on the site. Images of text and math are very strongly discouraged. Jul 18, 2022 at 17:13

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 $$e_\ell/2$$.
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.