Problem 7 · The late train
Motion: the delay per kilometre at reduced speed is constant.
Integer answer, at most 4 digitsA train service was disrupted by an accident after covering 30 km. The speed was reduced to four fifths of the original speed, causing a 45-minute delay. Had the accident happened 18 km further on, the train would have arrived 36 minutes late. What was the original speed in km/h?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: the delay is generated only on the stretch covered at reduced speed. Each kilometre at $\tfrac{4}{5}v$ takes $\tfrac{5}{4v}$ hours instead of $\tfrac{1}{v}$: an extra $\tfrac{1}{4v}$ h/km.
If $D$ is the total distance, the affected stretch is $D - 30$ km long (and $D - 48$ in the second scenario):
$$\frac{D-30}{4v} = \frac{45}{60} = \frac{3}{4}, \qquad \frac{D-48}{4v} = \frac{36}{60} = \frac{3}{5}.$$
Subtracting the two equations:
$$\frac{18}{4v} = \frac{3}{4} - \frac{3}{5} = \frac{3}{20} \;\Longrightarrow\; 4v = 120 \;\Longrightarrow\; v = 30\ \text{km/h}.$$
(Incidentally, $D = 120$ km.)
Answer: 30