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 reflects how fast vehicles are moving along a road segment by averaging over the space they occupy, which means weighting by how long each vehicle spends on the segment. It is defined as the total distance traveled by all vehicles divided by the total time they spend to traverse the segment. If every vehicle covers the same distance, the space-mean speed becomes the harmonic mean of their speeds. For example, with two vehicles moving the same distance D at speeds v1 and v2, SMS = 2D / (D/v1 + D/v2) = 2 / (1/v1 + 1/v2), which is the harmonic mean of v1 and v2. In general, SMS = N / sum(1/vi) for N vehicles, i.e., the harmonic mean of the speeds. This weighting is why the harmonic mean is used for space-mean speed: slower vehicles spend more time on the segment and thus have a larger influence on the average when you measure over space. The arithmetic mean simply averages speeds without accounting for how long each vehicle impacts the segment, the geometric mean isn’t tied to the distance-time relationship here, and a maximum speed is not an average at all.

Space-mean speed reflects how fast vehicles are moving along a road segment by averaging over the space they occupy, which means weighting by how long each vehicle spends on the segment. It is defined as the total distance traveled by all vehicles divided by the total time they spend to traverse the segment. If every vehicle covers the same distance, the space-mean speed becomes the harmonic mean of their speeds. For example, with two vehicles moving the same distance D at speeds v1 and v2, SMS = 2D / (D/v1 + D/v2) = 2 / (1/v1 + 1/v2), which is the harmonic mean of v1 and v2. In general, SMS = N / sum(1/vi) for N vehicles, i.e., the harmonic mean of the speeds.

This weighting is why the harmonic mean is used for space-mean speed: slower vehicles spend more time on the segment and thus have a larger influence on the average when you measure over space. The arithmetic mean simply averages speeds without accounting for how long each vehicle impacts the segment, the geometric mean isn’t tied to the distance-time relationship here, and a maximum speed is not an average at all.