What Direction Should a Man Row a Boat to Cross a River in the Shortest Time?

[20292]

Homework Statement

A man can row a boat at 4 m/s in still water. The current in a river is 2 m/s. The river is 40 m wide. If he wants to cross the river in the smallest possible time at what direction should he aim the boat?

Given:
V[B,E] = 4m/s
V[W,E] = 2 m/s

Down is negative.

Homework Equations

V[B,W] = V[B,E] + - V[W,E]

The Attempt at a Solution

V[B,E]:
-V[B,E] (Boat is going West in my drawing, so I made that negative)

V[W,E]:
-V[W,E] (Current is down... so negative)Then:

V[B,W] = -V[B,E] i + V[W,E] j (component form)

tan theta = -2/4

theta = 26.6 degrees.Nowhere in there, did I minimize time though. Where did I go wrong?

Thank you.

 

[WJSwanson]

Does the question specify whether he needs to cross so that he winds up directly across from where he started? Because if so, you would just find the angle such that (if you take "across" to be the positive x direction) the y-component of his velocity is the negative of the current's velocity.

If not, then I would think he should just point straight across the stream. (Remember, the current flows along the stream, but the path across it is perpendicular to the stream.)

 

[20292]

Does the question specify whether he needs to cross so that he winds up directly across from where he started? Because if so, you would just find the angle such that (if you take "across" to be the positive x direction) the y-component of his velocity is the negative of the current's velocity.

If not, then I would think he should just point straight across the stream. (Remember, the current flows along the stream, but the path across it is perpendicular to the stream.)

No, he can land anywhere on the other side. It doesn't need to be directly across from where he started.

 

[WJSwanson]

Well, then, since the current flows perpendicular to the boat's displacement, the best way to maximize the x-component of the boat's velocity would be to just aim it straight across, unless there's some detail I'm missing.

 

[Quantum Mind]

I am a learner too and I do not know if this is correct, but the shortest distance is vertically across. But since the current is flowing let us say due east with some velocity, should not the boat be aimed at some angle NW so that the net displacement is along due North, the shortest distance?

 

[WJSwanson]

The shortest total distance is directly across, yes. However, assuming that the distance across is the same everywhere, you minimize the time interval by maximizing the *vector component* in the direction of the opposite side of the river.

You can confirm this by calculating the time it takes if he aims the boat directly across versus if he aims it at the angle that counteracts the current.