Does the velocity of light change when it goes into a different medium?

Within a given medium (e.g. a vacuum, air, glass) the speed is fixed regardless of the wavelength (the frequency varies so the product of the frequency and the wavelength is a constant).

However the wavelength of a photon is dependent upon the medium through which it travels; the frequency remains a constant when a photo moves from one medium to another and therefore (in that sense) the velocity is dependent on the wavelength.

Sort of, and no... Sort of, if you meant velocity, no if you meant speed.

Using #E = hnu#, where #nu# is the frequency, and #nu = 1/lambda#...

Note that we can also write energy of light as:

#E = mc^2#

Thus:

#E = hnu = mc^2#

#mc = (hnu)/c = p#

the momentum of light. Since #lambda = c/(nu)#:

#mc = h/lambda#

#color(blue)(lambda = h/(mc))#

And so we have the equation for wavelength versus the speed of light.

The speed (scalar) of light is practically constant within a single media, yes, but the velocity (vector) of light is not. We can create translucent (semitransparent) surfaces that let through lower wavelengths of light due to their higher energy (that's how the Schott glass shortpass filter works, for example).

Note however, that these filters were developed in order to allow specific wavelengths to pass through. These were designed to alter the velocity of light by either reflecting it or letting it pass through, depending on the wavelength.

This effect of filtering is based on wavelength. The cause of this, specifically, is human being innovation in creating a special medium, rather than purely the wave-particle nature of light.

Another possible example is light scattering in the sky.

Since the index of refraction is wavelength-dependent, light scatters in the sky, and because of the following equation (https://en.wikipedia.org/wiki/Static_light_scattering#Theory),

where basically (and this was repeatedly mentioned by my Instrumental Design professor):
#I_"scatter" prop 1/lambda^4#

...the scattering intensity is higher for lower wavelengths, and blue light is the lowest visible wavelength, thus the sky is blue. If light were to not scatter ever, the sky would be rather white (and boring)!

Anyways, light scattering implies light changes direction, which by definition changes the velocity of light (but not its speed). In this case, the extent to which the velocity of light changes as it crosses into another medium is wavelength-dependent.

However, it is the crossover into different media that is the cause for the change in velocity occurring at all.