Q162
ArithmeticSpeed, Time & Distance
A vehicle covers half the distance with uniform speed $v_1$ and the other half with uniform speed $v_2$. The average speed of the vehicle is
- a.$\frac{v_1+v_2}{2}$
- b.$\frac{v_1+v_2}{v_1v_2}$
- c.$\frac{v_1v_2}{v_1+v_2}$
- d.$\frac{2v_1v_2}{v_1+v_2}$
Answer: (D) $\frac{2v_1v_2}{v_1+v_2}$
Let the total distance be $2d$. Time for the first half $=\frac{d}{v_1}$ and for the second half $=\frac{d}{v_2}$. Average speed $=\frac{\text{total distance}}{\text{total time}}=\frac{2d}{\frac{d}{v_1}+\frac{d}{v_2}}=\frac{2v_1v_2}{v_1+v_2}$, the harmonic mean of the two speeds. The arithmetic mean $\frac{v_1+v_2}{2}$ (option A) applies only when the two speeds are maintained for equal times, not equal distances. Example: 40 km/h going and 60 km/h returning gives 48 km/h, not 50 km/h.