Question

A bus covers half of the total distance with a speed of 30 kmh\(^{-1}\) and other half with a speed of 60 kmh\(^{-1}\). The average speed during the total journey is

• A. 35 kmh\(^{-1}\)
• B. 40 kmh\(^{-1}\)
• C. 45 kmh\(^{-1}\)
• D. 42 kmh\(^{-1}\)
• E. 50 kmh\(^{-1}\)

Hint:

Caution: Do not take the simple arithmetic mean \((30+60)/2 = 45\). This only works if the travel times are equal. For equal distances, always use the harmonic mean.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Average speed is defined as the total distance covered divided by the total time taken. When a journey is divided into two equal distances covered at different speeds, the average speed is the harmonic mean of the speeds.

Step 2: Key Formula or Approach:
If a body covers half distance at speed \(v_1\) and the other half at speed \(v_2\), then:
\[ V_{avg} = \frac{2 v_1 v_2}{v_1 + v_2} \]

Step 3: Detailed Explanation:
Given:
\(v_1 = 30 \text{ km/h}\)
\(v_2 = 60 \text{ km/h}\)
Applying the formula:
\[ V_{avg} = \frac{2 \times 30 \times 60}{30 + 60} \]
\[ V_{avg} = \frac{3600}{90} \]
\[ V_{avg} = 40 \text{ km/h} \]

Step 4: Final Answer:
The average speed during the total journey is 40 kmh\(^{-1}\).