Train delay

Due to a breakdown, the train spent 16 minutes standing on the track behind Brno. He "eliminated" this delay, so after the start, the 80 km-long section went at a speed 10 km/h higher than originally planned. What speed was it, and what was it supposed to be?

Correct answer:

v₁ =  50 km/h
v₂ =  60 km/h

Step-by-step explanation:

$$\begin{gathered} s=80\text{km} \\ {v}_0=10\text{km/h} \\ {t}_0=16\text{min}\to \text{h}=16:60\text{h}=0.26667\text{h} \\ s=v_1\cdot t \\ s=v_2\cdot (t-t_0) \\ {v}_2=v_1+v_0 \\ s=v_1\cdot t \\ s=v_2\cdot t-v_2\cdot t_0 \\ {v}_2=v_1+v_0 \\ s=(v_2-v_0)\cdot t \\ s=v_2\cdot t-v_2\cdot t_0 \\ s=v_2\cdot t-v_0\cdot t \\ s=v_2\cdot t-v_2\cdot t_0 \\ 80=v_1\cdot t \\ 80=v_2\cdot (t-16/60) \\ {v}_2=v_1+10 \\ 80=v_1\cdot t; \\ 80=(v_1+v_0)\cdot (t-16/60) \\ 80=v_1\cdot t+v_0\cdot t-16/60\cdot v_1-16/60\cdot v_0 \\ t=1.6\text{h} \\ {v}_1=s/t=80/1.6=50\text{km/h} \end{gathered}$$

$v_2=v_1+v_0=50+10=60\text{km/h}$

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