13.111. The train was delayed by $t$ hours. By increasing the speed by $m$ km/h, the driver eliminated the delay on a section of $s$ km. Determine what speed the train should have had on this section if there had been no delay.
Solution.

Let $m_{1}$ km/h be the speed of the train if it were to travel without delay; $t_{1}$ h be the travel time of the train in this case. By condition $m_{1} \cdot t_{1}=s$. Traveling faster at a speed of $m_{1}+m$ km/h, the train was on the road for $\left(t_{1}-t\right)$ h. Then $s=\left(m_{1}+m\right)\left(t_{1}-t\right)$. From this, we have $\left\{\begin{array}{l}m_{1} \cdot t_{1}=\left(m_{1}+m\right)\left(t_{1}-t\right), \\ t_{1}=\frac{s}{m_{1}} .\end{array}\right.$ Solving the system, we find $m_{1}=\frac{\sqrt{t m(4 s+t m)}-t m}{2 t}$.

Answer: $\frac{\sqrt{t m(4 s+t m)}-t m}{2 t} \mathrm{km} /$ h.