Question

Spot speeds of vehicles observed at a point on a highway are 40, 55, 60, 65 and 80 km/h. The space-mean speed (in km/h, round off to two decimal places) of the observed vehicles is \(\underline{\hspace{1cm}}\).

Hint:

The space-mean speed is calculated as the harmonic mean of the spot speeds, and is useful in traffic flow analysis.

Answer 1

Correct Answer: 55.5 - 58.5

The space-mean speed is the harmonic mean of the spot speeds. The formula for the space-mean speed is: \[ v_{\text{sm}} = \frac{n}{\sum_{i=1}^{n} \frac{1}{v_i}} \] Where: 
- \( n = 5 \) is the number of vehicles, 
- \( v_1, v_2, v_3, v_4, v_5 \) are the spot speeds of the vehicles. 
Substituting the values: \[ v_{\text{sm}} = \frac{5}{\frac{1}{40} + \frac{1}{55} + \frac{1}{60} + \frac{1}{65} + \frac{1}{80}} \] Calculating the sum: \[ v_{\text{sm}} = \frac{5}{0.025 + 0.01818 + 0.01667 + 0.01538 + 0.0125} = \frac{5}{0.08773} \approx 57.03 \, \text{km/h} \]