A canoe has a velocity of 0.470m/s southeast relative to the earth. The canoe is on a river that is flowing at 0.590 m/s east relative to the earth. Find the direction of the velocity of the canoe relative to the river.
The velocity of the canoe is 0.470 m/s south-east relative to the Earth. It is moving on a river that is flowing at 0.59 m/s towards the East.
The velocity of the canoe with respect to the Earth can be divided into a component towards the East equal to 0.47*sin 45 and a component towards the South equal to 0.47*cos 45. The velocity of the flow of the river is 0.59 m/s towards the East.
Relative to the river, the component of the velocity of the canoe towards the East is 0.47*cos 45 - 0.59 = -0.2576. This implies that the canoe is moving towards the West relative to the river at 0.2576 m/s. It also has a component towards the South equal to 0.3323 m/s.
The angle of the velocity of the canoe with respect to the river is an angle A south of west where tan A = 0.47*sin 45/0.3323
A = arc tan ((.59 - 0.47*cos 45)/(0.47*sin 45)) = 52.21 degrees.
The canoe is moving in a direction 52.21 degrees south of west with respect to the river.