With what speed should a galaxy move outward with respect to earth so that the sodium-D line at wavelength 5890 $$\mathring{A}$$ is observed at 5896 $$\mathring{A}$$?

This is an application of the Doppler effect for light (redshift). When a source moves away from the observer, the observed wavelength increases.

For the Doppler effect: $$\frac{\Delta\lambda}{\lambda} = \frac{v}{c}$$, where $$v$$ is the recession speed and $$c = 3 \times 10^5 \text{ km/s}$$.

Here $$\lambda = 5890$$ Å and $$\Delta\lambda = 5896 - 5890 = 6$$ Å.

So $$v = c \cdot \frac{\Delta\lambda}{\lambda} = 3 \times 10^5 \times \frac{6}{5890} = 3 \times 10^5 \times 1.02 \times 10^{-3} \approx 306 \text{ km/s}$$.

The galaxy moves outward at approximately $$306 \text{ km s}^{-1}$$.