Test whether a requested straight crossing is feasible

Begin with a measurement question

A velocity stated relative to moving water is not the same as velocity relative to the bank. Add vectors: boat relative to ground equals boat relative to water plus water relative to ground. The supplied virtual channel has width 12 m northward and a uniform current eastward. This is a mathematical exercise only, not swimming, boating or rescue advice.

Build the algebraic model

If the boat aims north relative to water at speed v, it crosses in 12/v seconds while drifting east by current speed times that time. It reaches the opposite bank but generally not the directly opposite point. Confusing those two goals creates an apparent contradiction between otherwise correct calculations.

Check a concrete implication

To follow a straight northward ground path, the boat must supply a westward relative component equal in magnitude to the eastward current u. Its remaining northward component is √(v²−u²). A finite crossing therefore requires v greater than u. At equality, all available relative speed is directed upstream and the northward component is zero; a zero denominator is not a finite crossing time.

Challenge the evidence boundary

For u=1 m/s and v=2 m/s, north-aimed relative motion takes 6 s and drifts 6 m east. A straight-across ground path instead uses northward speed√3 m/s, takes12/√3≈6.9282 s and points30° west of north relative to the water. The slower direct route follows a different heading constraint, not an error in velocity addition. Real currents, steering and safety are not modeled.

Worked original case

At current3 m/s and relative speed4 m/s, straight-across ground speed is√7 m/s and time ≈4.53557 s. At equal speeds 2 m/s there is no finite straight-across solution, although north-aimed relative motion still reaches the bank downstream.

Predict and investigate

Does relative speed equal to current suffice for a finite directly opposite crossing? Compare current 0,1,2 at relative speed 2, retaining north-aimed drift and straight-across feasibility separately. Predict the equality boundary before checking. Explain which goal remains possible when current exceeds relative speed.

Changed numerical practice

Try the changed case before feedback

For width 12 m,current3 m/s and relative speed5 m/s, find the straight-across crossing time in seconds.

Numerical result in the requested unitsCompare this result

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Reflect and source-check

Which velocity belongs in the crossing-time calculation for each heading constraint?

Original explanations, synthetic records and models; checked 30 October 2026 against OpenStax College Physics 2e. The guided calculation is algebra-based; no calculus prerequisite is assumed. All activity is virtual. No physical launch, drop, water crossing, machinery operation or medical decision is requested. A public answer or saved capture does not verify independent mastery or actual measurement technique.