Question

The spot speeds (in km/h) of eight vehicles in a traffic stream are 42, 52, 56, X, 53, 62, 65, and 48. X is the spot speed of the fourth vehicle. The Time Mean Speed of the traffic stream is 56.25 km/h. After determining the value of X, the calculated Space Mean Speed of the traffic stream is _________ km/h. (rounded off to two decimal places)

Hint:

The Time Mean Speed (TMS) is the average of the spot speeds, while the Space Mean Speed (SMS) considers the time spent by vehicles to cover the distance, and thus is usually lower than the TMS.

Answer 1

Step 1: Determine the value of X (the spot speed of the fourth vehicle) 
The Time Mean Speed (TMS) is given by the formula: \[ TMS = \frac{V_1 + V_2 + V_3 + \cdots + V_n}{n} \] where:
\( V_1, V_2, \dots, V_n \) are the spot speeds of the vehicles,
\( n \) is the number of vehicles.
Given that:
The Time Mean Speed (TMS) is 56.25 km/h,
The spot speeds are 42, 52, 56, \( X \), 53, 62, 65, 48.
We can use the formula for Time Mean Speed: \[ 56.25 = \frac{42 + 52 + 56 + X + 53 + 62 + 65 + 48}{8} \] Simplifying the equation: \[ 56.25 = \frac{378 + X}{8} \] Multiplying both sides by 8: \[ 450 = 378 + X \] Solving for \( X \): \[ X = 450 - 378 = 72 \] So, the spot speed of the fourth vehicle, \( X \), is 72 km/h. 
Step 2: Calculate the Space Mean Speed (SMS)
The Space Mean Speed (SMS) is given by the formula: \[ SMS = \frac{n}{\frac{1}{V_1} + \frac{1}{V_2} + \cdots + \frac{1}{V_n}} \] Substituting the values for the spot speeds (42, 52, 56, 72, 53, 62, 65, 48) into the formula: \[ SMS = \frac{8}{\frac{1}{42} + \frac{1}{52} + \frac{1}{56} + \frac{1}{72} + \frac{1}{53} + \frac{1}{62} + \frac{1}{65} + \frac{1}{48}} \] Calculating the reciprocals: \[ \frac{1}{42} = 0.02381, \quad \frac{1}{52} = 0.01923, \quad \frac{1}{56} = 0.01786 \] \[ \frac{1}{72} = 0.01389, \quad \frac{1}{53} = 0.01887, \quad \frac{1}{62} = 0.01613 \] \[ \frac{1}{65} = 0.01538, \quad \frac{1}{48} = 0.02083 \] Summing these reciprocals: \[ 0.02381 + 0.01923 + 0.01786 + 0.01389 + 0.01887 + 0.01613 + 0.01538 + 0.02083 = 0.13501 \] Now, calculating the Space Mean Speed: \[ SMS = \frac{8}{0.13501} \approx 59.26 \, {km/h} \] 
Conclusion: The Space Mean Speed (SMS) of the traffic stream is 59.26 km/h (rounded to two decimal places).