Let the speed of a boat in still water be \( u \) km/hr and the speed of the water current in a river be \( v \) km/hr. If a man rows distance \( D \) down stream and same upstream, then average speed of total journey is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{(u+v)(u-v)}{u} \)
The downstream speed is \( u+v \) and the upstream speed is \( u-v \). The total distance is 2D, and the total time is
\( \displaystyle \frac{D}{u+v}+\frac{D}{u-v} \)
\( \displaystyle =\frac{D\left[(u-v)+(u+v)\right]}{(u+v)(u-v)} \)
\( \displaystyle =\frac{2uD}{(u+v)(u-v)} \)
Average speed is total distance divided by total time, so
\( \displaystyle \text{average speed}=\frac{2D}{\frac{2uD}{(u+v)(u-v)}} \)
\( \displaystyle =\frac{(u+v)(u-v)}{u} \)
Why the other options are incorrect
Option (1) \( \displaystyle\frac{D}{u+v}+\frac{D}{u-v} \) — Incorrect. This is the total time of the journey, not the average speed.
Option (3) \( \displaystyle\frac{2D}{(u+v)(u-v)} \) — Incorrect. This is not total distance divided by total time; the total time is \( \frac{2uD}{(u+v)(u-v)} \).
Option (4) \( \displaystyle\frac{u+v}{u-v} \) — Incorrect. This is the ratio of the downstream and upstream speeds, not a speed.