Question

A boat capable of making 6.0 km/h in still water is used to cross a river flowing at speed of 4.0 km/h. The boat is directed (from the perpendicular to the shore) in such a way that its resulting motion is straight across the river.
At what angle (in ^o) must the boat be directed?

What is the resultant speed relative to the shore?

Answer 1

The water in the river is flowing downstream at 4.0 km/h. If you drove the boat perpendicular to the flow, you would float 4 meters downstream for every 6 meters you moved away from the bank. So you need to aim upriver slightly, so that you travel upstream somewhat, but you'll be carried by the water downstream, but you also travel away from the bank.
You just need to balance the upstream speed with the downstream speed, so that the boat will appear to go directly across the river. When you diagram the problem, you draw a right triangle with one leg equal to 4 and the hypotenuse equal to 6 because if the water wasn't flowing (like in a lake) the boat would travel 6 km/h at an angle (the hypotenuse) and 4 km/h parallel to the shore (opposite leg.) But since the flowing water moves the boat parallel to the shore at 4 km/h, in the opposite direction of the upstream component, they will cancel out, leaving the adjacent leg as the only motion of the boat, perpendicular to the shore.