Space-Mean Speed is the harmonic mean of speeds. Which option reflects this definition?

Space-Mean Speed is the harmonic mean of speeds. Which option reflects this definition?

• A. Arithmetic mean of speeds
• B. Geometric mean of speeds
• C. Maximum speed
• D. Harmonic mean of speeds

Answer: (D)

Space-Mean Speed measures how fast a whole set of vehicles covers a road section, so it must reflect how long each vehicle spends in that section. Imagine N vehicles each traveling the same length L. The total distance covered by all of them is NL. The time for the i-th vehicle is t_i = L / v_i, so the total time is sum(t_i) = L * sum(1/v_i). The space-mean speed is total distance divided by total time: SMS = (NL) / (L * sum(1/v_i)) = N / sum(1/v_i). That quantity is exactly the harmonic mean of the speeds, defined as H = N / sum(1/v_i). So space-mean speed equals the harmonic mean of the individual speeds. This differs from the arithmetic mean, which would treat all speeds equally without accounting for the fact slower vehicles spend more time in the section and thus contribute more to the overall timing. The harmonic mean correctly weights slower speeds more, which is why it describes SMS.

Space-Mean Speed measures how fast a whole set of vehicles covers a road section, so it must reflect how long each vehicle spends in that section. Imagine N vehicles each traveling the same length L. The total distance covered by all of them is NL. The time for the i-th vehicle is t_i = L / v_i, so the total time is sum(t_i) = L * sum(1/v_i). The space-mean speed is total distance divided by total time: SMS = (NL) / (L * sum(1/v_i)) = N / sum(1/v_i). That quantity is exactly the harmonic mean of the speeds, defined as H = N / sum(1/v_i). So space-mean speed equals the harmonic mean of the individual speeds.

This differs from the arithmetic mean, which would treat all speeds equally without accounting for the fact slower vehicles spend more time in the section and thus contribute more to the overall timing. The harmonic mean correctly weights slower speeds more, which is why it describes SMS.