The escape velocity from a spherical planet $$A$$ is $$10 km/s.$$ The escape velocity from another planet $$B$$ whose density and radius are 10% of those of planet $$A$$, is ______$$m/s.$$

We need to find the escape velocity from planet B given the escape velocity from planet A is 10 km/s.

Recall the escape velocity formula: $$v_e = \sqrt{\frac{2GM}{R}} = \sqrt{\frac{2G \cdot \frac{4}{3}\pi R^3 \rho}{R}} = R\sqrt{\frac{8\pi G\rho}{3}}$$ so $$v_e \propto R\sqrt{\rho}$$.

For planet B, density $$\rho_B = 0.1\rho_A$$ and radius $$R_B = 0.1R_A$$. Therefore,

$$\frac{v_{eB}}{v_{eA}} = \frac{R_B\sqrt{\rho_B}}{R_A\sqrt{\rho_A}} = 0.1 \times \sqrt{0.1} = \frac{1}{10} \times \frac{1}{\sqrt{10}} = \frac{1}{10\sqrt{10}}$$

It follows that $$v_{eB} = \frac{10}{10\sqrt{10}} = \frac{1}{\sqrt{10}} \text{ km/s}$$. Converting to m/s gives

$$v_{eB} = \frac{1000}{\sqrt{10}} = \frac{1000\sqrt{10}}{10} = 100\sqrt{10} \text{ m/s}$$

The escape velocity from planet B is $$100\sqrt{10}$$ m/s, which matches Option B. Therefore, the answer is Option B.