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A canoe has a velocity of 0.40m/s southeast relative to the earth. The canoe is on a river that is flowing 0.50m/s east relative to the earth. Find the velocity(magnitude and direction ) of the canoe relative to the river.

Discussion:

We apply the relative velocity relation. The relative velocities are $$\vec{v_{CE}}$$, the canoe relative to the earth, $$\vec{v_{RE}}$$, the velocity of the river with respect to Earth and $$\vec{v_{CR}}$$, the velocity of the canoe relative to the earth.
$$\vec{v_{CE}}=\vec{v_{CR}}+\vec{v_{RE}}$$
Hence $$\vec{v_{CR}}=\vec{v_{CE}}-\vec{v_{RE}}$$
The velocity components of $$\vec{v_{CR}}$$ are $$-0.5+\frac{0.4}{\sqrt{2}}=-0.217m/s$$( in the east direction)
Now, for the velocity component in the south direction
$$\frac{0.4}{\sqrt{2}}=0.28$$ (in the south direction)
Now, the magnitude of the velocity of canoe relative to river $$\sqrt{(-0.217)^2+(0.28)^2}=0.356m/s$$
If we consider $$\theta$$ as the angle between the canoe and the river,the direction of the canoe with respect to the river can be given by
$$\theta=52.5^\circ$$ ( in south west direction)