Following spot speed data were collected on a section of a road: 56 km/h, 68 km/h, 49 km/h, 58 km/h, 52 km/h, 66 km/h and 62 km/h. What is the space-mean-speed of the corresponding traffic stream?
- A58.7 km/h
- B59.8 km/h
- C58.0 km/hCorrect
- D56.3 km/h
Solution
Concept
Space-mean-speed is calculated as the total distance traveled divided by the total time taken by all vehicles, which mathematically equates to the harmonic mean of individual spot speeds.
Formula
Vs = n / Σ(1/vi)
Step-by-step solution
1. Identify the given spot speeds (vi): 56, 68, 49, 58, 52, 66, 62 km/h. 2. Count the number of observations (n): n = 7. 3. Calculate the sum of the reciprocals of the speeds: Σ(1/vi) = (1/56 + 1/68 + 1/49 + 1/58 + 1/52 + 1/66 + 1/62). 4. Σ(1/vi) ≈ (0.017857 + 0.014706 + 0.020408 + 0.017241 + 0.019231 + 0.015152 + 0.016129) = 0.120724. 5. Calculate Space Mean Speed (Vs) = n / Σ(1/vi) = 7 / 0.120724 ≈ 57.983 km/h. 6. Rounding to the nearest option gives 58.0 km/h.
Explanation
Space-mean-speed (Vs) is the harmonic mean of the spot speeds of individual vehicles. It represents the average speed of vehicles occupying a given length of a road lane at a particular instant, which is always less than or equal to the time-mean-speed (arithmetic mean).
Why the other options are wrong
A: This is the Time Mean Speed (arithmetic average of the speeds). B: This value is higher than the arithmetic mean, which is mathematically impossible for space mean speed. D: This value is significantly lower than the harmonic mean calculated.
Shortcut
Space mean speed is always lower than the arithmetic mean (Time mean speed). Here, Arithmetic Mean = 58.7 km/h, so the answer must be slightly less than 58.7.
Common mistake
Calculating the arithmetic mean (58.7 km/h) instead of the harmonic mean. The arithmetic mean is the 'Time Mean Speed'.
Correct answer: C) 58.0 km/h