Understanding Average Speed: The Case of a Two-Part Journey

When planning road trips or analyzing vehicle efficiency, one fundamental question often arises: what is the average speed for a journey? A perfect example to illustrate this concept is a car that travels 150 km at 75 km/h, then continues for another 200 km at 100 km/h. Let’s explore how to calculate the average speed for the entire trip and why it differs from a simple mean of the two speeds.

Understanding the Context

What Is Average Speed?

Average speed is defined as the total distance traveled divided by the total time taken. Unlike average velocity, which involves direction, average speed is purely a measure of how quickly a journey is completed in terms of distance over time.

The Journey Breakdown

A car travels 150 km at a speed of 75 km/h, then another 200 km at 100 km/h. What is the average speed for the entire trip?

Key Insights

  • Segment 1:
    Distance = 150 km
    Speed = 75 km/h
    Time = Distance ÷ Speed = 150 km ÷ 75 km/h = 2 hours

  • Segment 2:
    Distance = 200 km
    Speed = 100 km/h
    Time = 200 km ÷ 100 km/h = 2 hours

Total Distance and Total Time

  • Total Distance = 150 km + 200 km = 350 km
  • Total Time = 2 hours + 2 hours = 4 hours

Continue Reading

Find the minimum value of \((\cos x + \sec x)^2 + (\sin x + \csc x)^2\).
To find the minimum value of the expression \((\cos x + \sec x)^2 + (\sin x + \csc x)^2\), let us start by rewriting \(\sec x\) and \(\csc x\) in terms of \(\cos x\) and \(\sin x\):
Let \(a = \cos x\) and \(b = \sin x\). Then:

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Final Thoughts

Calculating Average Speed

Average Speed = Total Distance ÷ Total Time
= 350 km ÷ 4 h = 87.5 km/h

Why Not Just Average of 75 and 100?

While it might be tempting to compute the average as (75 + 100) ÷ 2 = 87.5, this works only if both segments take equal time—which is not the case here. Since the car spends more time at the slower speed (2 hours at 75 km/h vs. 2 hours at 100 km/h), the actual average speed is influenced by the longer duration at lower speed. This example highlights an essential insight: average speed is distance-weighted, not speed-averaged.

Practical Implications

Understanding average speed helps drivers estimate arrival times, compare routes, and assess fuel efficiency. In this trip, even though the average speed is 87.5 km/h (heavier than the maximum speed of 100 km/h), the fast segment compensates for the slower one, resulting in overall efficiency.