Given, diameter of spherical ball $\left(d\right)=1 \mathrm{cm}=0.01\mathrm{ m}$

Radius of spherical ball $\left(r\right)=\frac{0.01}{2}\mathrm{ m}$

Density of spherical ball $\left(\rho \right)=5\times {10}^3\mathrm{ kg} {\mathrm{m}}^{-3}$

Density of liquid $\left(\sigma \right)=3\times {10}^3\mathrm{ kg}{\mathrm{ m}}^{-3}$

Coefficient of viscosity $\left(\eta \right)=0.1 \mathrm{N}-\mathrm{s} {\mathrm{m}}^{-2}$

We know that,

Terminal velocity is given by

$v=\frac{2}{9}\frac{g{r}^2\left(\rho -\sigma \right)}{\eta }$

$=\frac{2}{9}\times 10\times {\left(\frac{0.01}{2}\right)}^2\times \frac{\left(5\times {10}^3-3\times {10}^3\right)}{0.1} m {\mathrm{s}}^{-1}$

$v=\frac{10}{9} m {\mathrm{s}}^{-1}$

Distance travelled in one second after attaining terminal velocity$=\left(\frac{10}{9}\times 1\right)\mathrm{ m}$

$=\frac{10}{9}\mathrm{ m}$