A nucleus disintegrates into two nuclear parts, in such a way that ratio of their nuclear sizes is $$1 : 2^{1/3}$$. Their respective speed have a ratio of $$n : 1$$. The value of $$n$$ is _______

Given: Ratio of nuclear sizes (radii) = $$1 : 2^{1/3}$$.

Since nuclear radius $$r \propto A^{1/3}$$ (where A is mass number):

$$\frac{r_1}{r_2} = \frac{A_1^{1/3}}{A_2^{1/3}} = \frac{1}{2^{1/3}}$$

Therefore $$\frac{A_1}{A_2} = \frac{1}{2}$$

By conservation of linear momentum (initial nucleus at rest):

$$A_1 v_1 = A_2 v_2$$

$$\frac{v_1}{v_2} = \frac{A_2}{A_1} = 2$$

The ratio of speeds is $$n : 1 = 2 : 1$$, so $$n = 2$$.