A railway train with a total weight of 300 tons can reach a maximum speed of $35 \mathrm{~km}$ per hour on a horizontal track. Calculate the power of the locomotive if the coefficient of friction is 0.005. What is the maximum speed this train can achieve on a track with a gradient of $1: 100$ (the tangent of the slope angle $=0.01$)?
Let's denote the total weight of the train by $m$, its speed in meters per second by $v$, the angle of inclination of the slope by $\alpha$, the coefficient of friction by $\varepsilon$, the force overcoming obstacles by $P_{1}$, and the number of horsepower by $H_{p}$. Since on a horizontal track the obstacle is only represented by friction, which is the product of the mass $(m)$ and the coefficient of friction $(\varepsilon)$, we have

$$
P_{1}=m \varepsilon
$$

Furthermore, the power in horsepower is obtained by dividing the work done per second and expressed in meter-kilograms by 75, so

$$
H_{p}=\frac{P_{1} \cdot v}{75}=\frac{m \cdot \varepsilon \cdot v}{75}
$$

Given $m=300$ tons, $v=\frac{35000}{3600} \frac{\mathrm{m}}{\mathrm{sec}} ; \varepsilon=0.005$, substituting these values we get

$$
H_{p}=194.4 \text { horsepower. }
$$

When the train is moving uphill on a slope, it has to overcome two forces: one is the force of gravity $\left(P_{2}\right)$, the other is friction $\left(P_{1}\right)$; however,

$$
\begin{gathered}
P_{2}=m \sin \alpha \\
P_{1}=m \cos \alpha \cdot \varepsilon
\end{gathered}
$$

and thus

$$
P=P_{1}+P_{2}=m(\varepsilon \cos \alpha+\sin \alpha)
$$

The power of the train is always expressed by the equation

$$
H_{p}=\frac{P \cdot w}{75}
$$

where $w$ is the speed of the train in meters per second.

If $H_{p}=194.4$ horsepower $=194.4 \frac{\mathrm{m} \cdot \mathrm{kg}}{\mathrm{sec}}$ and we substitute the value of $P$ into this equation, we get

$$
w=3.24 \frac{\mathrm{m}}{\mathrm{sec}}
$$

or 11.6 km per hour. This will be the maximum speed of the train. The calculation shows that due to the smallness of the angle $\alpha$, $\cos \alpha=1, \sin \alpha=0$ can be taken with negligible error.

(Mihály Devecis.)

The problem was also solved by: Erdős A., Groffits G., Goldziher K., Kármán T., Kornis Ö.