The Average-Speed Paradox

Settled by elementary kinematics (harmonic mean for equal-distance legs).

This is one of the most common 'gotcha' problems in introductory kinematics and appears in countless popular-physics collections; Lange places it in his Kinematics chapter as a direct illustration of the arithmetic-vs-harmonic-mean confusion. The trap is powerful because the arithmetic mean feels obviously correct and the error persists even among people who know the formula for average speed (total distance / total time) if they don't pause to apply it carefully.

A one-line question quietly separates people who compute an average from people who understand what 'average' means physically.

Explanations that circulated

• Assuming that because there are exactly two speeds, the average speed must be their arithmetic mean (v1+v2)/2 — a formula that is only valid when EQUAL TIME is spent at each speed.
• Failing to notice that when equal DISTANCES (not equal times) are covered at each speed, more time is actually spent at the slower speed, so the true average speed is the harmonic mean, 2v1v2/(v1+v2) = 40 km/h in this example, not 45 km/h.

References

• Fundamentals of Physics, David Halliday, Robert Resnick, Jaren Walker, Wiley, 2013
• Wikipedia article (verified live, fetched during research), Wikipedia, 2026