When anyone expedites anything, he or she is actually "freeing" the feet for faster action. An expedition is a group of people sent on some important undertaking and so their feet must be free to move without hindrance. Impede means "to tangle the feet" or "to obstruct" the movement of the feet.
People often see this ped element in other words. When people refer to "pedal extremities", they mean "feet". When anyone pushes the pedals of a bicycle, it is done with the feet. A pedestrian must use the feet for walking. A quadruped has four feet while a centipede has "100 feet"; or a large number of them because it may be impossible to count all of them.
While pes may refer to a foot as a measure of length, it can also refer to what the Roman poet, Horace, spoke humorously of as, sesquipedalia verba (words which are a "foot and a half long") and which exist in English as sesquipedalians with the same meaning.
Keep in mind that all of the ped words which you see in English are not from the Latin "foot" or "feet". There are also some Greek ped words in English which do not mean "foot"!
The word carpet, for example, ultimately derives from the Latin carpo, which meant to "pluck" or to "card" wool, and it is believed that the first carpets were of wooly cloth made of unravelled threads.
Then there is the term scarce, which English inherited from the French escars, "scanty", originally from the Latin ex, "out", and carpo, "pluck". It's like "plucking" from the cookie jar until the cookies become "scanty" and scarce.
Another related word is excerpt, from Latin excerptus (ex, "out" and carpo, "pluck") which refers to something that has been "plucked out" of its context.
The result is that the idea of "plucking" streams through the three widely divergent words just as a scarce thread of color can be woven through the carpet with which this excerpt started.
These basic words and their related forms can be seen in this carpo-, carp- (cerp-) unit of "to pluck, to pick out, to gather, to select" words.