Energy consumption and speed If we want two bodies A and B of same mass to move at constant speeds $v_a$ and $v_b$ , across a surface that offers frictional force of $f$, then the external force required in both cases would be the same. (This external force = frictional force $f$). And if they are made to move the same distance $x$, then the work done and the energy expended by the external agent would be the same too $(=fx)$, even though the velocities are different.
So why do physics teachers say travelling at twice the speed expends four times the energy (using the expression for kinetic energy)?
Are they talking about producing different accelerations across the same distance in two bodies of the same mass?
 A: One object that is travelling at twice the speed of another has four times as much kinetic energy, provided the two objects have the same mass. Of course, if the second faster object has only one quarter of the mass of the first slower object then they have the same kinetic energy, and so on.
You can think of kinetic energy as the energy that must be expended to get a stationary object moving at a certain speed in the absence of friction. If the object has mass $m$ and we exert a constant force $F$ over a distance $s$ then the energy expended is $Fs$. If the object's final speed is $v$ then
$\displaystyle v^2 = 2as = \frac {2Fs} m
 \\ \displaystyle \Rightarrow Fs = \frac 1 2 mv^2$
which is the expression for kinetic energy.
In your example we are not measuring kinetic energy, but we are measuring the energy that must be expended on each object to keep it travelling at a constant speed despite friction. If the friction force $f$ is the same for both objects and if they both travel the same distance $x$ then the energy required to keep them moving at a constant speed over this distance is the same for both objects - it is $fx$. Notice that this conclusion  holds even if the objects have different masses and different speeds - so it has nothing to do with the kinetic energy of the objects (except in so far as the energy expended here is replacing the kinetic energy of the objects that is converted to heat due to friction).
