Question

A car travels half the distance with a velocity of $20 \text{ kmh}^{-1}$ and another half distance with a velocity of $30 \text{ kmh}^{-1}$ along a straight road. The average velocity of the car in $\text{km h}^{-1}$ is

• A. 35
• B. 25
• C. 48
• D. 50
• E. 24

Hint:

Never take the simple arithmetic mean (which would be 25) for equal-distance problems. The car spends more time traveling at the slower speed, so the average velocity will always be closer to the lower speed.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
Average velocity is total distance divided by total time. When a journey is split into two equal distance halves traveled at different speeds, the average speed is the harmonic mean of the two speeds.
Step 2: Key Formula or Approach:
For equal distances \(d\) traveled at speeds \(v_1\) and \(v_2\), the average velocity \(v_{\text{avg}}\) is:
\[ v_{\text{avg}} = \frac{2v_1 v_2}{v_1 + v_2} \] Step 3: Detailed Explanation:
Let the total distance be \(2d\).
Time taken for the first half: \(t_1 = \frac{d}{v_1} = \frac{d}{20}\).
Time taken for the second half: \(t_2 = \frac{d}{v_2} = \frac{d}{30}\).
Total time: \(T = t_1 + t_2 = \frac{d}{20} + \frac{d}{30} = d\left(\frac{3 + 2}{60}\right) = \frac{5d}{60} = \frac{d}{12}\).
Average velocity:
\[ v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{d/12} = 2d \times \frac{12}{d} = 24\text{ kmh}^{-1} \] Using the direct formula:
\[ v_{\text{avg}} = \frac{2(20)(30)}{20 + 30} = \frac{1200}{50} = 24\text{ kmh}^{-1} \] Step 4: Final Answer:
The average velocity is 24.