Determining the Speed of a Passenger Train Based on Its Overtaking Time

A goods train, traveling at a constant speed, leaves a station one hour before a passenger train. The passenger train catches up with the goods train after spending four hours on the track. This text outlines the steps to calculate the speed of the passenger train using basic principles of speed, distance, and time.

Introduction to the Problem

Let's examine the problem where a goods train initially leaves a station at a speed of 45 kilometers per hour (km/h). After one hour, a passenger train starts and catches up with the goods train in four hours. The objective is to determine the speed of the passenger train.

Step-by-Step Solution

Step 1: Calculate the Distance Traveled by the Goods Train

Since the goods train travels at 45 km/h and starts one hour before the passenger train, by the time the passenger train overtakes it, it has been traveling for 5 hours.

Distance Speed times; Time Distance 45 km/h times; 5 hours 225 km

Step 2: Use the Distance to Find the Speed of the Passenger Train

The passenger train takes four hours to cover the same distance of 225 km. To find the speed of the passenger train, we use the formula:

Speed Distance / Time Speed 225 km / 4 hours 56.25 km/h

Alternative Methods to Confirm the Result

The results can be verified using different approaches:

Method 1: Direct Distance Calculation

The total journey covered by the goods train up to the point of overtaking is 225 km. Therefore, the speed of the passenger train is:

Speed 225 km / 4 hours 56.25 km/h

Method 2: Distance and Speed Equations

Let the distance covered by the goods train up to the time it is overtaken be x. From the time equation:

x/45 - 4 1 x/45 5 x 225 km

Using the speed formula again:

Speed Distance / Time Speed 225 km / 4 hours 56.25 km/h

Method 3: Complex Calculation with Similar Triangles

Initially stating the speed of the goods train as 22.5 km/h and using the equation for distance:

Time for goods train 5 hours, speed 45 km/h rarr; Distance 45 times; 5 225 km

For the passenger train, distance 60 km/h times; 3 hours 180 km. Since both trains cover the same distance:

Speed of goods train 180 km / 8 hours 22.5 km/h

Conclusion

The speed of the passenger train is determined as 56.25 km/h based on the given conditions. This example illustrates the practical application of speed, distance, and time relationships in solving real-world problems using basic principles.