Maybe I am missing something but whenever I calculate the maximum speed attainable by a train knowing only its power and mass I always get values that make no sense. If someone could explain my error that would be greatly appreciated. As I understand the relevant formulas are as follows. I am assuming constant mass for simplicity and ignoring the need to accelerate. All I am interested in is the maximum sustainable speed, analogous to terminal velocity.
$$F_\text{tractive effort}=(M_\text{total} \cdot F_g) \cdot C_\text{friction}$$
$$P=F \cdot v$$
$$\therefore P=((M_\text{total} \cdot F_g) \cdot C_F) \cdot v$$
From my reading, a reasonable coefficient of kinetic friction for steel wheels on steel track is $C_\text{friction}=0.5$
So if we try to calculate the speed of the heaviest train ever run we have the following numbers.
Sources: https://en.wikipedia.org/wiki/Longest_trains,
https://en.wikipedia.org/wiki/GE_AC6000CW
$$M_\text{total}=99734\text{t}$$
$$P=8 \cdot 3500\text{kW}=28000\text{kW}$$
$$\therefore v=\frac{28000000}{99743000\cdot 9.81\cdot 0.5}\approx 0.0572\text{m}/\text{s}$$
For reference, according to google a garden snail moves at $0.013\text{m}/\text{s}$
Now clearly my result of point o' 6 meters per second is ridiculous for the speed of even a freight train.
Any insight would be much appreciated.
Best Answer
The estimation for the coefficient of rolling friction of $C_{\rm friction} = 0.5$ sounds unrealistic. The whole point of a train is that it rolls with very little friction. The above value looks more like the coefficient of sliding friction.
For example if you use a more realistic $C_{\rm friction} = 0.0018$ the resulting rolling resistance force is
$$F_{\rm friction} = C_{\rm friction} M_{\rm total} g = 1760.5\,\mathrm{kN}$$
and the top speed
$$ v_{\rm max} = \frac{P_{\rm total}}{F_{\rm friction}} = 15.9\,\mathrm{m\, s^{-1}}$$