A) \[{{2}^{1/3}}:1\]
B) \[1:{{3}^{1/2}}\]
C) \[{{3}^{1/2}}:1\]
D) \[1:{{2}^{1/3}}\]
Correct Answer: D
Solution :
Law of conservation of momentum gives \[{{m}_{1}}{{v}_{1}}={{m}_{2}}{{v}_{2}}\] \[\Rightarrow \] \[\frac{{{m}_{1}}}{{{m}_{2}}}=\frac{{{v}_{2}}}{{{v}_{1}}}\] But \[m=\frac{4}{3}\pi {{r}^{3}}\rho \]or\[m\propto {{r}^{3}}\] \[\therefore \] \[\frac{{{m}_{1}}}{{{m}_{2}}}=\frac{r_{1}^{3}}{r_{2}^{3}}=\frac{{{v}_{2}}}{{{v}_{1}}}\] \[\Rightarrow \] \[\frac{{{r}_{1}}}{{{r}_{2}}}={{\left( \frac{1}{2} \right)}^{1/3}}\] \[\left( \because \frac{{{v}_{1}}}{{{v}_{2}}}=\frac{2}{1} \right)\] \[\therefore \] \[{{r}_{1}}:{{r}_{2}}=1:{{2}^{1/3}}\]
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