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River-boat problems
Intermediate
Resultant velocity of boat is vector sum of boat and current velocity.
The boat's ground velocity is the vector sum of its velocity relative to water and the river current. To cross in the shortest time, point the boat perpendicular to the bank; to land directly opposite, aim upstream so the current is cancelled: sinθ = vr/vb.
Boat speed 5 m/s, river 3 m/s, width 100 m: shortest time 20 s, drift 60 m.
Shortest time and shortest path are different strategies.
In more detail
A boat on a river has velocity relative to the water plus the water's velocity relative to the bank. To cross in the shortest time, the boat must head across the stream, but it will then drift downstream.
Real-world uses
- Ferries aim upstream to compensate for the current and reach a point directly opposite.
- Kayakers plan routes using relative velocity to reach a landing point.
- A ferry crossing a river must head upstream at an angle to reach a point directly opposite, because the current carries it sideways.
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Frequently asked questions
How do I cross in the shortest time?
Point the boat directly across the river, perpendicular to the bank, which gives the minimum crossing time.
How do I reach the point directly opposite?
Head upstream at an angle so the downstream drift is cancelled by the current.
What heading gives the shortest crossing time?
Aim straight across the river, perpendicular to the bank. The current then only moves the boat sideways, not along the path across.
Related topics in Kinematics
Velocity at any point of projectile · Effect of air resistance · Relative velocity in 2D · Rain and umbrella problem · Uniform circular motion · Angular displacement and velocity