A man wishing to cross a river flowing with velocity u jumps at an angle $\theta$ with the river flow. a. Find the net velocity of the man with respect to the ground if he can swim with speed v in still water. b. In what direction does the boat actually move? c. Find how far from the point directly opposite to the starting point does the boat reach the opposite bank, if the width of the river is $d$.

Resolving the swimmer's velocity v into components, the component perpendicular to the river flow is v sin(theta), and the component parallel to the flow is v cos(theta). The net velocity of the man relative to the ground is the vector sum of the river's velocity u and his swimming velocity, giving horizontal and vertical components of (u + v cos(theta)) and v sin(theta), respectively. The time taken to cross the river of width d is found by dividing the width by the vertical component, yielding t = d / (v sin(theta)). To find the downstream drift from the point directly opposite, multiply the horizontal net velocity by this crossing time: (u + v cos(theta)) * d / (v sin(theta)). The result is =(u+v\cos\theta)\dfrac{d}{v\sin\theta}.

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