Does cassette tape change its velocity while playing?
The details of how a cassette drive works are well covered by this Wikipedia article. The tape is pulled by a capstan next to the playback head, and this capstan pulls the tape at a steady rate.



(picture from the Wikipedia article)

You probably need to click on the picture to see it full size. I have indicated the capstan by a red arrow. The take-up spool doesn't rotate at a fixed speed. It uses a slipping drive, as badjohn says in his answer, so it takes up the tape at the speed the capstan moves it.