I apologise for what is probably merely stupidity on my part, but has been vexing me for rather a while, and no one I ask can provide any insight. A body undergoes two accelerations. The first is from 0 to 10 m/s, the second from 10 to 20 m/s. For simplicity's sake I will say F= 1N, a= 1 m/s2 and m = 1kg, for both of the accelerations. So in both cases, it takes 10 seconds for the particle to accelerate, and so if it were powered by some kind of engine, the time it is running would also be 10 seconds. As the force it produces is constant, the amount of fuel burnt in each case would logically be the same. But apparently it's not. Here is the problem: The distance travelled in the first instance is 50m, with u=0, v=10, a=1. As work done= Fs, the energy transferred is therefore 50J. So 50 joules worth of fuel is burnt. For the second acceleration, u=10 and v=20, so the distance travelled is 150m. Hence the energy transferred the second time is 150J. So despite the engine providing the same amount of force for the same time, it has used three times as much fuel. Unless it knows how fast it is going (with relation to what, anyway?), how could it take more energy to accelerate the second time? Thank you for you patience and (hopefully) your help.