Explanations:The concept of average speed is crucial in understanding the relationship between distance, speed, and time. When a car travels equal distances at different speeds, the average speed is not simply the arithmetic mean of the speeds. Instead, it is calculated using the harmonic mean of the speeds. This is because the time taken to travel each segment is inversely proportional to the speed, and the harmonic mean accounts for this relationship.
The formula for the harmonic mean of speeds \( s_1, s_2, \ldots, s_n \) is given by:
\[
H = \frac{n}{\frac{1}{s_1} + \frac{1}{s_2} + \cdots + \frac{1}{s_n}}
\]
where \( n \) is the number of speeds. In this problem, the car travels four equal distances at speeds of 2 kmph, 3 kmph, 4 kmph, and 6 kmph. To find the average speed, we apply the harmonic mean formula:
\[
H = \frac{4}{\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{6}}
\]
First, we calculate the sum of the reciprocals of the speeds:
\[
\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{6} = \frac{6}{12} + \frac{4}{12} + \frac{3}{12} + \frac{2}{12} = \frac{15}{12} = \frac{5}{4}
\]
Next, we substitute this sum back into the harmonic mean formula:
\[
H = \frac{4}{\frac{5}{4}} = 4 \times \frac{4}{5} = \frac{16}{5} = 3.2 \text{ kmph}
\]
Thus, the average speed of the car is 3.2 kmph. This calculation demonstrates the importance of using the harmonic mean when dealing with equal distances and different speeds, as the arithmetic mean would not accurately represent the average speed in this scenario.