Calculating Average Speed: A Case Study of an Intercity Express Train

Inter-city travel often involves varying speeds to accommodate different terrains, traffic conditions, and operational logistics. This article delves into a practical example where an intercity express train travels at different speeds over different distances and calculates the average speed for the entire journey. This case study is useful for students and professionals in transportation and logistics to understand the concept of average speed in practical scenarios.

Introduction to the Problem

Consider an intercity express train that travels a distance of D at varying speeds: one-fourth of the distance at 80 km/h and the remaining distance at 60 km/h. Our goal is to calculate the average speed of the train for the entire journey. This problem is a classic example in transportation engineering and can help in optimizing routes and schedules in real-world scenarios.

Breakdown of the Journey

To break down the journey into manageable segments, let's assume the total distance of the train's journey is D.

Distance at 80 km/h: One-fourth of the total distance, which is frac{1}{4}D. Distance at 60 km/h: The remaining three-fourths of the total distance, which is frac{3}{4}D.

Time Calculation for Each Segment

Next, let's calculate the time taken for each segment of the journey.

Time at 80 km/h: The time taken to cover frac{1}{4}D at a speed of 80 km/h is calculated as follows:

text{Time}_1 frac{frac{1}{4}D}{80} frac{D}{320} hours.

Time at 60 km/h: The time taken to cover the remaining frac{3}{4}D at a speed of 60 km/h is calculated as follows:

text{Time}_2 frac{frac{3}{4}D}{60} frac{3D}{240} frac{D}{80} hours.

Total Time for the Journey

The total time for the journey is the sum of the times taken for each segment:

text{Total Time} text{Time}_1 text{Time}_2 frac{D}{320} frac{D}{80}.

To add these fractions, we need a common denominator. The least common multiple (LCM) of 320 and 80 is 320. Thus:

frac{D}{80} frac{4D}{320}.

So, the total time:

text{Total Time} frac{D}{320} frac{4D}{320} frac{5D}{320} frac{D}{64} hours.

Calculating the Average Speed

Finally, the average speed of the train can be calculated using the formula for average speed, which is the total distance divided by the total time:

text{Average Speed} frac{D}{frac{D}{64}} 64 text{ km/h}.

Therefore, the average speed of the train for the entire distance is 64 km/h.

Advanced Problem Solution

Let's consider another scenario where the journey is divided into three parts with different speeds: one-fourth at 80 km/h, one-third at 60 km/h, and the rest at 100 km/h. We will solve this problem in a similar manner:

Let D denote the total distance in km. The distances for the three parts are as follows:

One-fourth of the distance: frac{1}{4}D covers at 80 km/h. One-third of the distance: frac{1}{3}D covers at 60 km/h. The rest of the distance: frac{1}{12}D covers at 100 km/h (since frac{3}{12}D frac{4}{12}D frac{5}{12}D 12/12 1).

The time taken for each segment is calculated as follows:

Time at 80 km/h: text{Time}_1 frac{frac{1}{4}D}{80} frac{D}{320} hours. Time at 60 km/h: text{Time}_2 frac{frac{1}{3}D}{60} frac{D}{180} hours. Time at 100 km/h: text{Time}_3 frac{frac{5}{12}D}{100} frac{D}{240} hours.

The total time for the journey is the sum of these times:

text{Total Time} frac{D}{320} frac{D}{180} frac{D}{240}.

To add these fractions, we find a common denominator. The LCM of 320, 180, and 240 is 2400. Thus:

frac{D}{320} frac{7.5D}{2400}, frac{D}{180} frac{13.333D}{2400}, frac{D}{240} frac{10D}{2400}.

So, the total time:

text{Total Time} frac{7.5D}{2400} frac{13.333D}{2400} frac{10D}{2400} frac{30.833D}{2400} frac{5.139D}{400}.

The average speed is then calculated as:

text{Average Speed} frac{D}{frac{5.139D}{400}} 77.84 text{ km/h}.

Therefore, the average speed for the entire journey is 77.84 km/h.

Conclusion

The calculation of average speed is crucial in transportation and logistics to optimize travel efficiency. Understanding how to break down distances and calculate times for each segment can help in planning efficient routes and schedules.