Boat crossing a river – relative velocity
PhysicsKinematicsAges 15–16
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Sign in to playA boat moves at speed v₁₂ relative to the water on a river flowing at v₂₃. Change the heading angle θ with the slider or by dragging on the river to see the velocity triangle v₁₃ = v₁₂ + v₂₃, the path, the crossing time and the drift; a graph shows how time and drift depend on θ. One button works out the upstream angle needed to land directly opposite.
Lesson: Relative velocity – adding velocities
What it shows
A boat on a river combines two velocities: its velocity relative to the water, v₁₂, set by its engine and heading, and the water's velocity relative to the bank, v₂₃. Its velocity relative to the bank is the vector sum v₁₃ = v₁₂ + v₂₃. Only the part of v₁₂ straight across the river carries the boat to the other side, so the crossing time is t = d/(v₁₂ cos θ), while the current pushes it sideways, giving a drift x = (v₂₃ − v₁₂ sin θ)·t. The model keeps the boat's speed and heading constant and ignores wind, waves and acceleration.
How to use
Set Boat speed relative to water, Current speed and River width, then press Start. Use Heading angle θ, or drag on the river, to aim the bow, and compare the crossing time and the landing point C in the readout. Press Head straight across for the fastest crossing and Land opposite for the angle that reaches B. Switch Current to Slow at banks, fast in middle to see a curved path.
Parameters you can change
- River width 20–300 m
- Boat speed relative to water 0.5–8 m/s
- Current speed 0–6 m/s
- Heading angle (positive: upstream) -85–85 °
- Current Uniform across the river, Slow at banks, fast in middle
Questions to explore
- Why does a faster current not change the crossing time when the boat heads straight across?
- How must the boat be aimed to land directly opposite, and what happens to the crossing time?
- If the current is faster than the boat, can it ever land directly opposite its starting point?
