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 formula for Time Mean Speed (TMS) is: \[ TMS = \frac{V_1 + V_2 + V_3 + \cdots + V_n}{n} \]. Here, \( V_1, V_2, \dots, V_n \) represent the spot speeds of the vehicles, and \( n \) is the total number of vehicles.
Given: TMS = 56.25 km/h. The spot speeds are 42, 52, 56, \( X \), 53, 62, 65, 48 (n=8).
Applying the TMS formula: \[ 56.25 = \frac{42 + 52 + 56 + X + 53 + 62 + 65 + 48}{8} \].
Simplifying the sum of known speeds: \[ 56.25 = \frac{378 + X}{8} \].
Multiply both sides by 8: \[ 450 = 378 + X \].
Solve for \( X \): \[ X = 450 - 378 = 72 \].
Therefore, the spot speed of the fourth vehicle, \( X \), is 72 km/h.
Step 2: Calculate the Space Mean Speed (SMS)
The formula for Space Mean Speed (SMS) is: \[ SMS = \frac{n}{\frac{1}{V_1} + \frac{1}{V_2} + \cdots + \frac{1}{V_n}} \].
Substitute the spot speeds (42, 52, 56, 72, 53, 62, 65, 48) into the formula:
\[ SMS = \frac{8}{\frac{1}{42} + \frac{1}{52} + \frac{1}{56} + \frac{1}{72} + \frac{1}{53} + \frac{1}{62} + \frac{1}{65} + \frac{1}{48}} \].
Calculate the reciprocals of the spot speeds:
\[ \frac{1}{42} \approx 0.02381, \quad \frac{1}{52} \approx 0.01923, \quad \frac{1}{56} \approx 0.01786 \]
\[ \frac{1}{72} \approx 0.01389, \quad \frac{1}{53} \approx 0.01887, \quad \frac{1}{62} \approx 0.01613 \]
\[ \frac{1}{65} \approx 0.01538, \quad \frac{1}{48} \approx 0.02083 \]
Sum the reciprocals: \[ 0.02381 + 0.01923 + 0.01786 + 0.01389 + 0.01887 + 0.01613 + 0.01538 + 0.02083 \approx 0.13501 \].
Calculate SMS: \[ SMS = \frac{8}{0.13501} \approx 59.26 \, {km/h} \].
Conclusion: The Space Mean Speed (SMS) of the traffic stream is approximately 59.26 km/h (rounded to two decimal places).