Question

A vehicle travels half of the total distance (L) with speed $V_1$ and the other half with speed $V_2$, then its average speed over this whole distance is:

• A. $(V_1+V_2)/2$
• B. $(2V_1+V_2)/(V_1+V_2)$
• C. $2V_1V_2/(V_1+V_2)$
• D. $L(V_1+V_2)/V_1V_2$

Hint:

If distance halves are equal, average speed is the Harmonic Mean. If time halves are equal, it's the Arithmetic Mean.

Answer 1

The Correct Option is C

Step 1: Concept
Average speed is defined as $\frac{\text{Total Distance}}{\text{Total Time}}$.

Step 2: Time Calculation
Time for the first half: $t_{1} = \frac{L/2}{V_1}$.
Time for the second half: $t_{2} = \frac{L/2}{V_2}$.

Step 3: Analysis
Total Time $T = \frac{L}{2V_1} + \frac{L}{2V_2} = \frac{L(V_1+V_2)}{2V_1V_2}$.
Average Speed $V_{avg} = \frac{L}{T} = \frac{L \times 2V_1V_2}{L(V_1+V_2)}$.

Step 4: Conclusion
The formula simplifies to the harmonic mean: $V_{avg} = \frac{2V_1V_2}{V_1+V_2}$. Final Answer: (C)