River Boat Problem: Minimizing Travel Time

[this_is_harsh]

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

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?

 

[TSny]

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 $\beta$.

 

[kuruman]

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"?