The concept used to solve this problem is the relative velocity.
Initially, resolve the velocities to determine the relative velocity in the horizontal and vertical directions.
Then, find the angle between the resultant of relative velocities and its horizontal component to determine the direction of velocity of the canoe relative to the river.
The relative velocity is the vector difference between the velocities of two bodies. Consider two bodies X and Y moving relative to each other. The relative velocity is the velocity that the body X would appear to an observer on the body Y and vice versa.
The expression for relative velocity is,
Here, is the relative velocity of the body X with respect to Y, is the velocity of the body X, and is the velocity of the body Y.
In the figure below, is the velocity of the canoe relative to earth and is the velocity of the river relative to earth.
Horizontal component of is,
Substitute for .
Vertical component of is,
Substitute for .
Horizontal component of is,
Vertical component of is,
The expression for relative velocity is,
The expression for relative velocity in the horizontal direction is,
Substitute for and for .
The expression for relative velocity in the vertical direction is,
Substitute for and for .
In the figure below, is the velocity of the canoe relative to the river. The angle gives the direction of .
The expression for tangent of the angle is,
The expression for the angle is,
Substitute for and for .
From the figure, the direction of is south of west.
Ans:The direction of the velocity of the canoe relative to the river is south of west.
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