Understanding Average Speed in a Multi-Segment Journey

When faced with complex journey problems, it's essential to break them down into manageable parts. In this article, we'll explore the problem of a person traveling a total distance of 100 km, broken into segments at different speeds. We'll walk through the process of calculating the average speed for the entire journey.

Total Distance and Speed Calculations

Alan covered a total distance of 100 km. He traveled the first 40 km at a speed of 60 km/h and the remainder of the distance (60 km) at the same speed.

Time Calculation for the First Segment

To calculate the time taken for the first segment, we use the formula:

Time Distance / Speed

For the first 40 km:

Time for first 40 km 40 km / 60 km/h 2/3 hours ≈ 0.67 hours

Time Calculation for the Second Segment

For the second segment, the remaining 60 km:

Time for remaining 60 km 60 km / 60 km/h 1 hour

Total Time Calculation

Adding the times from both segments:

Total Time Time for first 40 km Time for remaining 60 km 2/3 hours 1 hour 5/3 hours ≈ 1.67 hours

Average Speed Calculation

Finally, we calculate the average speed:

Average Speed Total Distance / Total Time 100 km / 5/3 hours 100 km * 3/5 60 km/h

Thus, Alan's average speed for the entire journey is 60 km/h.

Another Example

In another scenario, if we have a journey where the first half is covered at 40 km/h and the second half at 60 km/h, we can follow a similar method:

Time Calculation for the First Half

Time for first half Distance/Speed 100km / 80 km/h 5/4 hours

Time Calculation for the Second Half

Time for second half Distance/Speed 100km / 120 km/h 5/6 hours

Total Time Calculation

Total Time 5/4 5/6 25/12 hours

Average Speed Calculation

Average Speed Total Distance / Total Time 100 km / 25/12 48 km/h

So, the average speed for this journey is 48 km/h.

A Real-World Scenario

Let's use a more complex example with a total distance of 250 km, where the speed varies but the average speed is an important consideration:

Total distance 250 km

Breaking it down into segments:

Distance (km)Speed (km/h)Time (hours) 10050100/50 2 hours 12054120/54 20/9 hours 306030/60 1/2 hours

Total Time Calculation

Total time 2 20/9 1/2 53/9 hours

Average Speed Calculation

Average Speed Total Distance / Total Time 250 km / 53/9 4500/53 km/h ≈ 84.91 km/h

The average speed for this scenario is approximately 84.91 km/h.

Conclusion

The key to understanding average speed in multi-segment journeys is to break down the journey into its segments, calculate the time taken for each, and then use the total distance and total time to find the average speed.

By following these steps, you can solve any journey problem where the average speed is required.