Does Accelerating to Double the Speed Require Quadruple the Fuel Consumption, or Double the Fuel Consumption? I'm writing a piece on the efficiency of trains versus trucks, and one of the points I was making is that trains can take advantage of not having to stop and start multiple times. This is obviously beneficial, although probably negligible for highways vs cross-country trips when compared to rolling/wind resistance. But it made me think of something else that I can't get out of my head. If an object, let's say a train, is traveling twice as fast, did it require four times as much chemical energy, from fuel, to be used?
The actual numbers are irrelevant, but imagine two fully loaded trains of equal design and weight, let's say ten million kilograms. They both start from stop, but train A accelerates to 100 kmph, while train B accelerates to 200 kmph. Train B now has four times the kinetic energy, so did it burn four times the fuel? If force is mass*acceleration, and the force is generated by combustion, then it seems all we need to burn is the same rate of fuel, but for twice as long. Working backwards from momentum we see that we only needed twice the fuel, but working backwards from kinetic energy we need four times the fuel.
This same principle applies for every method of transportation. Two rockets ships of equal weight lead to the same confusing answer provided one of them applies the same thrust, and the same fuel burn rate, for simply twice as long. We have consumed twice the chemical energy, yet have four times the kinetic energy. Acceleration by propeller is no different.
In these examples I assumed no rolling/wind resistance in order to make things more clear. Obviously in real life we have quadratic wind/rolling resistance to overcome. But I'm still curious.
To make a long story short, does accelerating to twice the speed require four times the fuel, or twice the fuel?
I looked back through my physics 12 high school curriculum and couldn't find the answer. Then I sat down with some extended family, including an engineer, and asked them. No one could answer the question and we were all stumped, thinking that we must be missing something obvious.
 A: 
To make a long story short, does accelerating to twice the speed require four times the fuel, or twice the fuel?

It depends.
A force on the vehicle implies an interaction with something else. A train pushes against the tracks, a car pushes against the road, a propeller pushes against the fluid, a rocket pushes against the exhaust. You must consider that thing as well. Both momentum and energy are conserved, but only when you consider both the vehicle and the thing it pushes against.
So, suppose that an engine provides work $W$ while accelerating a vehicle of mass $m$ from $v_i$ to $v_f$ by pushing against something of mass $M$ which accelerates from $V_i$ to $V_f$.
Conservation of momentum gives $$m v_i+M V_i=m v_f+M V_f$$
Conservation of energy gives $$\frac{1}{2}m v_i^2 + \frac{1}{2}M V_i^2 + W=\frac{1}{2}m v_f^2 + \frac{1}{2}M V_f^2$$
These two equations assume that the “push” mass has one velocity at the end and that the mass of the vehicle is constant. Things like rockets violate both of those conditions, so it would just be an approximation. However, you can make an “instantaneous” version of these that is accurate for rockets too. But for this question we will just approximate the rocket using these equations.
For the car $v_i=V_i=0$ and then we can eliminate $V_f$ and solve for $W$ to get $$W=\frac{(m+M)}{2M}m v_f^2 \approx \frac{1}{2}m v_f^2$$ where the approximation is in the limit where $M$ is much larger than $m$. Here, doubling $v_f$ quadruples $W$.
For the rocket if we start with the rocket at rest then again $v_i=V_i=0$ and $V_f=-V_e$ where $V_e$ is the exhaust velocity of the rocket engine. Then we can eliminate $M$ and solve for $W$ to get $$W=\frac{1}{2} m v_f (V_e+v_f) \approx \frac{1}{2} m v_f V_e$$ where the approximation is in the limit where $V_e$ is much larger than $v_f$. Here doubling $v_f$ doubles $W$

we were all stumped, thinking that we must be missing something obvious.

Usually people miss conservation of momentum. You didn’t seem to miss momentum, but perhaps you neglected that it is conserved. Or perhaps you neglected the energy that goes into the thing the vehicle pushes against
