Do electrons deccelerate through a resistor? In a circle the voltage drops across a resistor. This means that some electrons lost some of their electrostatic potential energy. Where does that energy go, and how? For potential energy to be lost work must be done opposite to the conservative force causing the potential energy. This implies that there is an acceleration – so my question is: Do electrons speed up or slow down because of a resistor?
Alternatively,
 E = resistivity x current density, meaning a higher resistivity implies a stronger electric field. The difference in electric field strength resulting from the difference in resisitivities causes a net force, meaning there is acceleration shortly, until the dragging force becomes equal and a new drift velocity is achieved. The drift velocity in a resistor is greater than in the surrounding wires.
 A: Electrons inside solids have a very random motion. They keep on bumping around here and there. What happens when you put a battery? The motion of these electrons is only slightly disturbed. Free electrons inside the metal move at large speeds, a battery sets up an electric field inside the solid which tries to push the one side but the acceleration caused is very low. This is why electrons do not move like uniform particles, line by line behind each other in a queue but rather they just slightly drift towards one side while still doing that random motion.This means there are a large number of collisions.The electric fields energy is not going completely in pushing but alot is going into making them collide, this generates heat and this is how energy is lost and potential drops, it runs off as HEAT ENERGY.
We cannot say if electrons slow down or speed up or since they are moving at random speeds, but we can definitely say that the rate at which the move a little side by side along the solid becomes less. This rate is called DRIFT VELOCITY and it becomes less.
If a current of I amperes flows through a resistor of resistance R in voltage V then the amount of heat dissipated or potential energy lost per second is given by:
$$ I^2R = V^2/R = VI $$
For more detail you can study about these on wikipedia:
$$ $$
->Drift velocity
$$ $$
->Heat loss in resistors
$$ $$
Also here's a helpful diagram to understand the motion of electrons
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A: There is a lot of random thermal motion of charges, but on average their speed (drift velocity) through a resistor is constant. Around a given series circuit the current is constant with respect to position, so the average speed depends only on the product of the cross sectional area and the free electron density. Since that is constant across a resistor the drift velocity is constant with respect to position also. 

For potential energy to be lost work must be done opposite to the conservative force causing the potential energy. This implies that there is an acceleration

This is faulty logic. Suppose there are two forces, the conservative force from the electric field $F_E$ and non conservative resistive force $F_R$. Since, as you say, on average the work done by $F_R$ is equal and opposite that done by $F_E$ we have:
$$F_R d =-F_E d$$
$$F_R =-F_E$$
So on average the forces are equal and opposite which means that the average acceleration is 0. 
The drift velocity is given by: $$V_d=\frac{I}{e N A}$$ so it is possible for the drift velocity to change around a circuit even with a fixed current $I$, but it is not due to changes in resistivity. It is due to changes in the cross sectional area $A$ or the density of charge carriers $N$. Both $A$ and $N$ are independent of resistivity. It is clear that $A$ is independent, but an examination of tables of resistivity and charge carrier density shows that $N$ is also independent of resistivity. For example Mercury has a relatively high resistivity and moderate $N$, while Aluminum has a high $N$ but a low resistivity.
A: In the stationary state, the current along the whole circuit is the same, due to conservation of charge. To put it simply, take any tiny region of the circuit, and at any given time, in the stationary state, the number of electrons flowing in and out of that region must be the same.
Extending this to the whole circuit, the intensity is the same everywhere, so for wires of the same cross-section, no matter the resistivity of the material, the drift velocity must stay the same. 
This means there is no net drift acceleration inside a resistor. If there was, electrons would bunch up inside it, because more would be flowing in than flowing out, and charge up like a capacitor would, but you can check that doesn't happen.
The difference in potential energy across a resistor is macroscopically turned into heat, and is not spent on accelerating the electrons themselves.
A: The answer to the above questions is that the drift velocity of electrons must change for the current to remain the same if other factors change. For example, if the wire suddenly becomes thinner, like a bottle neck then the A in $$I=eNV_dA$$    becomes smaller and thus $V_d$ must increase. The drift velocity is also proportional on the relaxation time, so if that changes, the drift velocity must also change, all other things equal. In conclusion, the drift velocity inside a circuit will change depending on the components of the circuit.  Because the electrons change their drift velocity, it is plausible to conclude that there is some net acceleration/deceleration through the components of the circuit, all so as to keep the current itself constant throughout the circuit since charge MUST be conserved. Axiomatically, the important statement is that $I$ is constant before and after the resistor.
