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Considering the "water analogy" for electricity, it seems voltage is sort of like gravity:

"Electricity Flows like Water"

(image source: http://learn.olympiacircuits.com/electricity-flows-like-water.html)

Now when water actually falls like the illustration shows, the particles gain speed as their height decreases. Do electrons do the same thing, at least in a vacuum?

What about in a wire? Do the electrons reach some sort of "terminal velocity" so as not to be traveling faster at one end of a circuit than the other? What determines their travel speed?

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the particles gain speed as their height decreases. Do electrons do the same thing, at least in a vacuum?

Yes, an electron in a vacuum (say, in a vacuum tube) will continue to accelerate as long as it's under the influence of an electric field.

What about in a wire? Do the electrons reach some sort of "terminal velocity" so as not to be traveling faster at one end of a circuit than the other?

In a wire or other conductive material, though, electrons tend to interact ("bounce off") other particles (atomic nuclei, phonons, or other electrons) frequently, so their average speed tends to even out along the path of conduction.

What determines their travel speed?

For more details, research Drude theory.

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To make the water analogy work it is best to use a closed pipe. If water flows downward in a closed pipe it does not accelerate - if it did, the water at the bottom of the pipe would be moving faster than the water at the top of the pipe and therefore the bits of water would have to separate from one another creating a vacuum between the bits of water.
The effect of gravity on the water, rather than causing it to accelerate as it moves downward, is to transfer the force (via fluid pressure, or suction, if you want to call it that) to other portions of the fluid that might be upstream or downstream. Likewise, in a series circuit current does not flow faster through small resistances and slower through large resistances - all resistors receive the same current (coulombs per second, which is the equivalent of kg per second of water). But each resistor contributed toward determining what that one flow rate would be.

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Assume a current I = 1 ampere, and a wire of 2 mm diameter (radius = 0.001 m). This wire has a cross sectional area of 3.14×10−6 m2 (A = π × (0.001 m)2). The charge of one electron is q = −1.6×10−19 C. The drift velocity therefore can be calculated: enter image description here

Codus to Wikipedia.

A termal velocity can only be the speed of light, so let's calculate how much current and voltage we need to get to 0.75c:

I = ((8,5*10^28m^-3)*(3,14*10^-6m^2)*(1,6*10^-19C)(0,75c))

I = 9,60175284 × 10^12 A

To drive that current through a 1m * 2 mm^2 copper wire you need a voltage off

U = 2 * (0,0171 Ohm) * (9,60175284 × 10^12 A)

U = 3,2 *10^11 V

Almost enough to let the insulation of air breakdown from earth to the moon (if there were air in between).

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