A stone tied to a string of length $L$ is swirled in a verticle circle with the other end of the string at the center. At a certain instant of time, the stone is at its lowest position and has a speed $u$. Find the magnitude of the change in its velocity as it reaches a position, where the string is horizontal.

let the direction of counter clockwise initial velocity

$=\mathrm{ui}$

i is the directional of initial velocity

when the string is horizontal the body is covered the quarter of revolution the velocity is $v j$

$j$ is the direction of velocity towards y axis

change in the velocity is vj - ui

magnitude of the change in velocity $\sqrt{\mathrm{v}^{2}-\mathrm{u}^{2}}$

by using the conservation of energy

energy of the body is $=1 / 2 \mathrm{mu}^{2}$

when it goes to quarter position KE spent is doing work mgh and remaing KE

$\begin{array}{l} =1 / 2 m v^{2} \\ 1 / 2 m u^{2}=1 / 2 \mathrm{mv}^{2}+\mathrm{mgh} \\ \mathrm{h}=\mathrm{l} \\ 1 / 2 \mathrm{mu}^{2}=1 / 2 \mathrm{mv}^{2}+\mathrm{mgh} \\ \mathrm{v}^{2}-\mathrm{u}^{2}=-2 \mathrm{gl} \\ \mathrm{v}^{2}-\mathrm{u}^{2}+2 \mathrm{u}^{2}=2 \mathrm{u}^{2}-2 \mathrm{gl} \\ \mathrm{v}^{2}-\mathrm{u}^{2}+2 \mathrm{u}^{2}=2 \mathrm{u}^{2}-\mathrm{gl} \\ =\sqrt{\mathrm{v}^{2}+\mathrm{u}^{2}}=\sqrt{2}\left(\mathrm{u}^{2}-\mathrm{gl}\right) \end{array}$