🚣 River–Boat (Relative Velocity)

📖 The Physics

The boat's velocity relative to the ground is the vector sum of its velocity relative to the water (v_b, the way it is steered) and the water's velocity relative to the ground (v_r, the current). How you aim the boat decides the trade-off between speed and drift.

Resultant (vector sum):

\vec{V} = \vec{v_b} + \vec{v_r}

1) Shortest time — aim straight across (⟂ to the bank):

t = \dfrac{d}{v_b} \qquad x = \dfrac{v_r\, d}{v_b} \qquad V = \sqrt{v_b^{2} + v_r^{2}}

2) Shortest path — aim upstream so the drift is zero (needs v_b > v_r):

\sin\theta = \dfrac{v_r}{v_b} \qquad t = \dfrac{d}{\sqrt{v_b^{2} - v_r^{2}}}

3) 45° to the bank — aim upstream at 45°:

V_y = v_b\sin 45^\circ \qquad V_x = v_r - v_b\cos 45^\circ \qquad t = \dfrac{d}{V_y}

Notice case 1 gives the smallest timebut the boat lands downstream, while case 2 gives the smallest distance(straight across) but takes longer. Case 2 only works when v_b > v_r — otherwise the current always wins.

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