Two trains in pursuit

Statement

A regional train leaves a station at $80$ km/h. Half an hour later an Intercity leaves along the same route at $120$ km/h (multiple tracks, no collisions). After how long — and at what distance from the station — does the Intercity catch up with the regional train?

Solution

Unknown. $t=$ time of the Intercity since departure, in hours.

Equation. At that moment the regional train has been travelling for $t+0{,}5$ hours. The two distances covered must coincide: $120\,t = 80\,(t+0{,}5) \implies 120t-80t=40 \implies 40t=40 \implies t=\boxed{1}\text{ h}$ The Intercity catches the regional train $1$ h after its own departure, at $120$ km from the station. Check: the regional train, having left half an hour earlier, has travelled $1{,}5$ h, and $80\cdot 1{,}5=120$ km. Consistent.

Connections

Topics: First-degree equations
Concepts: Equation · Unknown
Methods: Pursuit motion problem
Skills: Modelling · Solving equations
Exercise type: Word problem