Let's analyze the scenario step-by-step.
-
Determine the distance traveled by the first train before the high-speed train departs: The first train leaves the station and travels for 1.5 hours before the second train starts. Its speed is 110 kilometers per hour.
\[ \text{Distance} = \text{Speed} \times \text{Time} = 110 \text{ km/h} \times 1.5 \text{ h} = 165 \text{ km} \]
-
Set up the equations to find when the high-speed train catches up: After 1.5 hours, the first train is already 165 kilometers ahead. The high-speed train now starts traveling at a speed of 260 kilometers per hour.
Let's denote \( t \) as the time in hours that the high-speed train travels until it catches up with the first train.
- In that time \( t \), the first train will continue to travel, and we need to express the total distance for both trains at that point.
The distance covered by the first train after the high-speed train starts traveling is: \[ \text{Distance}_{\text{first train}} = 110 \text{ km/h} \times (t + 1.5) = 110t + 165 \text{ km} \]
The distance covered by the high-speed train during that time is: \[ \text{Distance}_{\text{high-speed train}} = 260 \text{ km/h} \times t = 260t \text{ km} \]
-
Set the distances equal to each other: For the high-speed train to catch up, the distances must be equal: \[ 110t + 165 = 260t \]
-
Solve for \( t \): We subtract \( 110t \) from both sides: \[ 165 = 260t - 110t \] \[ 165 = 150t \] Now, divide both sides by 150: \[ t = \frac{165}{150} = 1.1 \text{ hours} \]
-
Final Answer: It will take the high-speed train 1.1 hours to catch up to the first train.