Boats & Streams

At a glance 10 formulas

1. 1Downstream and upstream$D = b + s,\quad U = b - s$
2. 2Boat and stream back$b = \dfrac{D + U}{2}$$s = \dfrac{D - U}{2}$
3. 3Round-trip time$T = \dfrac{d}{b + s} + \dfrac{d}{b - s} = \dfrac{2bd}{b^2 - s^2}$
4. 4Round-trip average speed$\bar{S} = \dfrac{2DU}{D + U} = \dfrac{b^2 - s^2}{b}$
5. 5Extra time upstream$d = \dfrac{(b^2 - s^2)\,\Delta t}{2s}$
6. 6Speed ratio$\dfrac{D}{U} = \dfrac{t_{\text{up}}}{t_{\text{down}}}\ \text{(same stretch)}$$\dfrac{D}{U} = \dfrac{d_{\text{down}}}{d_{\text{up}}}\ \text{(same time)}$
7. 7Upstream time ratio$t_{\text{up}} = n\, t_{\text{down}} \Rightarrow b : s = (n + 1) : (n - 1)$
8. 8Two boats meetingBoats heading towards each other close the gap at $b_1 + b_2$: the stream speeds one up and slows the other by the same amount, so it cancels.
9. 9Alternating legsAn hour downstream then an hour upstream leaves the boat $(b + s) - (b - s) = 2s$ km further downstream after each pair.
10. 10Something floatingA log or a dropped bottle drifts at $s$. Against the water the boat moves at $b$ both ways, so it gets back to the bottle in the time it spent going away.