Question

The travel times of three vehicles on a 2 km stretch of road are 4, 5, and 8 minutes. Assuming the speed of each vehicle to be constant in this stretch, the Space Mean Speed (in km/h) of the vehicles is (rounded off to two decimal places).

Hint:

Space Mean Speed is the harmonic mean of speeds, equal to total distance over total time.

Answer 1

Correct Answer: 21.18

Step 1: Recall what Space Mean Speed means.
Space Mean Speed (SMS) is the average speed of all vehicles that occupy a given length of road at a given instant. It equals the total distance travelled by all vehicles divided by the total time they take, which is the same as the harmonic mean of their individual speeds. This is different from Time Mean Speed, which is the plain arithmetic average of speeds recorded at a point.

Step 2: Convert each travel time to a speed.
Each vehicle covers the same distance, \(d = 2\) km.
\[ v_1 = \frac{d}{t_1} = \frac{2}{4/60} = \frac{2 \times 60}{4} = 30 \ \text{km/h} \]
\[ v_2 = \frac{d}{t_2} = \frac{2}{5/60} = \frac{2 \times 60}{5} = 24 \ \text{km/h} \]
\[ v_3 = \frac{d}{t_3} = \frac{2}{8/60} = \frac{2 \times 60}{8} = 15 \ \text{km/h} \]

Step 3: Apply the Space Mean Speed formula.
For \(n\) vehicles each covering the same distance,
\[ SMS = \frac{n}{\sum \frac{1}{v_i}} = \frac{n \, d}{\sum t_i} \]
The total distance travelled by all three vehicles is \(3 \times 2 = 6\) km. The total time taken is
\[ \sum t_i = 4 + 5 + 8 = 17 \ \text{min} = \frac{17}{60} \ \text{h} \]

Step 4: Compute the Space Mean Speed.
\[ SMS = \frac{6}{17/60} = \frac{6 \times 60}{17} = \frac{360}{17} = 21.176 \ \text{km/h} \]

Final Answer:
Rounded to two decimal places, the Space Mean Speed is 21.18 km/h.
\[ \boxed{SMS = 21.18 \ \text{km/h}} \]