Two steel spheres approach each other head-on with the same speed and collide elastically. After the collision one of the sphere's of radius r comes to rest, the radius of other sphere is

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$$\begin{gathered} \mathrm{Suppose} {\mathrm{m}}_1 \mathrm{and} {\mathrm{m}}_2 \mathrm{are} \mathrm{the} \mathrm{masses},\mathrm{velocity} \mathrm{before} \mathrm{collision} \mathrm{is} \mathrm{v} \mathrm{and} \mathrm{velocity} \\ \mathrm{of} \mathrm{the} \mathrm{steel} \mathrm{spheres} \mathrm{and} \mathrm{after} \mathrm{collision} \mathrm{the} \mathrm{volocity} \mathrm{of} \mathrm{the} \mathrm{second} \mathrm{sphere} \mathrm{is} \mathrm{v}\text{'}. \\ \mathrm{Then} \mathrm{from} \mathrm{the} \mathrm{conservation} \mathrm{of} \mathrm{momentum} \mathrm{and} \mathrm{conservation} \mathrm{of} \mathrm{energy} \mathrm{we} \mathrm{get} \\ \frac{1}{2}{\mathrm{m}}_1{\mathrm{v}}^2+\frac{1}{2}{\mathrm{m}}_2{\mathrm{v}}^2=\frac{1}{2}{\mathrm{m}}_2\mathrm{v}{\text{'}}^2................................. \left(1\right) \\ {\mathrm{m}}_1\mathrm{v}-{\mathrm{m}}_2\mathrm{v}={\mathrm{m}}_2\mathrm{v}\text{'} \\ \Rightarrow \mathrm{v}\text{'}=\frac{{\mathrm{m}}_1-{\mathrm{m}}_2}{{\mathrm{m}}_2}...............................\left(2\right) \\ \mathrm{putting} \mathrm{value} \mathrm{of} \mathrm{v}\text{'} \mathrm{on} \mathrm{equation} \left(1\right) \mathrm{we} \mathrm{will} \mathrm{get} \\ \frac{1}{2}{\mathrm{m}}_1{\mathrm{v}}^2+\frac{1}{2}{\mathrm{m}}_2{\mathrm{v}}^2=\frac{1}{2}{\mathrm{m}}_2{\left(\frac{{\mathrm{m}}_1-{\mathrm{m}}_2}{{\mathrm{m}}_2}\right)}^2{\mathrm{v}}^2 \\ {\mathrm{m}}_1+{\mathrm{m}}_2=\frac{{\left({\mathrm{m}}_1-{\mathrm{m}}_2\right)}^2}{{\mathrm{m}}_2} \\ {\mathrm{m}}_1{\mathrm{m}}_2+{{\mathrm{m}}_2}^2={{\mathrm{m}}_1}^2+{{\mathrm{m}}_2}^2-2{\mathrm{m}}_1{\mathrm{m}}_2 \\ 3{\mathrm{m}}_1{\mathrm{m}}_2={{\mathrm{m}}_1}^2 \\ {\mathrm{m}}_2=\frac{{\mathrm{m}}_1}{3}..................................\left(3\right) \\ \mathrm{So} \mathrm{mass} \mathrm{of} \mathrm{the} \mathrm{second} \mathrm{ball} \mathrm{is} \frac{1}{3} \mathrm{of} \mathrm{the} \mathrm{first} \mathrm{ball}. \mathrm{Since} \mathrm{they} \mathrm{are} \mathrm{made} \mathrm{by} \mathrm{same} \\ \mathrm{material}, \mathrm{then} \mathrm{the} \mathrm{volume} \mathrm{of} \mathrm{the} \mathrm{second} \mathrm{ball} \mathrm{will} \mathrm{aslo} \mathrm{be} \frac{1}{3}. \mathrm{So} \mathrm{if} \mathrm{volume} \mathrm{of} \mathrm{the} \\ \mathrm{first} \mathrm{still} \mathrm{ball} \mathrm{is} \mathrm{V} \mathrm{then} \mathrm{second} \mathrm{one} \mathrm{will} \mathrm{be} \frac{\mathrm{V}}{3} \\ \mathrm{Now} \mathrm{we} \mathrm{have} \\ \frac{4}{3}{\mathrm{\pi r}}^3=\mathrm{V}..................................\left(4\right) \\ \mathrm{and} \mathrm{if} \mathrm{r}\text{'} \mathrm{is} \mathrm{the} \mathrm{radius} \mathrm{of} \mathrm{the} \mathrm{second} \mathrm{ball} \mathrm{then} \mathrm{we} \mathrm{have} \\ \frac{4}{3}\mathrm{\pi r}{\text{'}}^3=\frac{\mathrm{V}}{3}...................................\left(5\right) \\ \mathrm{dividing} \left(5\right) \mathrm{by} \left(4\right) \mathrm{we} \mathrm{get} \\ \frac{\mathrm{r}{\text{'}}^3}{{\mathrm{r}}^3}=\frac{1}{3} \\ \mathrm{r}\text{'}=\frac{\mathrm{r}}{\sqrt[3]{3}} \end{gathered}$$

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