River Boat Problem: Minimizing Travel Time

AI Thread Summary
To minimize travel time across a river, the angle of rowing must be optimized, leading to the conclusion that θ should be 0 degrees and β less than 90 degrees. However, this creates a contradiction as a 0-degree angle results in a net velocity that does not align with the intended direction. The expression for velocity, v_A sin β, is not maximized at β = 90 degrees, indicating that v_A varies with β. The necessity to minimize travel time arises from the presence of a current, which alters the optimal angle compared to a scenario without current. Understanding these dynamics is crucial for effective navigation in flowing water.
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Homework Statement
Find minimum time required by the boat(v-a) to cross the river of width d , in which stream is moving with the velocity (v-b)
Relevant Equations
Time= distance_y / v_y
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From equation (3.7) , we know t = d/v_y = d / v_ABcosθ = d/ v_Asinβ
Now for time to be minimum , denominator must be maximum this implies θ=0 and β=90, but this doesn't make sense as when we try to row the boat at 0 degree with y-axis due to stream it will have a some net velocity which will lie in first quadrant , this clearly implies
β must be less than 90 degree, the what about the value of β we get above?
 
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The expression ##v_A \sin \beta## is not maximized for ##\beta = 90^o##. Note that ##v_A## cannot be treated as a constant when varying ##\beta##.
 
Why is there need to minimize? Suppose there is no current. The minimum crossing time is if the boat is aimed straight across from O to Q. Why would this change if a current is "turned on"?
 
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