A man wants to reach point $B$ on the opposite bank of a river flowing at a speed as shown in figure. What minimum speed relative to water should the man have so that he can reach point $B$ 

Let v be the speed of boatman in still water. 

Resultant of v and u should be along AB. Components of vb (absolute velocity of boatman) along x and y directions are,

vx=u-vsinθ and uy=vcosθ Further, tan45°=vyvx 1=vcosθu-vsinθ v=usinθ+cosθ =u2 sin(θ+450)

v is minimum at, 

θ+45°=90° θ=45° and vmin=u2