A canoe has a velocity of 0.400 southeast relative to the earth. The canoe is on a river that is flowing at 0.450 east relative to the earth.
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To determine the velocity of the canoe relative to the Earth, you need to combine the velocities of the canoe and the river.
Given:
Velocity of the canoe = 0.400 southeast
Velocity of the river = 0.450 east
Step 1: Convert the southeast velocity of the canoe into its components along the east and north directions.
Since southeast is a combination of east and south, we can break it down into two perpendicular components. Let's assume a right-angled coordinate system where east is the x-axis and north is the y-axis.
To convert the southeast velocity of the canoe into its east (x) and north (y) components, we need to calculate the magnitudes of the vertical and horizontal sides of the right triangle formed by the southeast vector:
sin(45°) = vertical side / hypotenuse
=> vertical side = hypotenuse * sin(45°) = 0.400 * sin(45°)
cos(45°) = horizontal side / hypotenuse
=> horizontal side = hypotenuse * cos(45°) = 0.400 * cos(45°)
Step 2: Calculate the resultant velocity of the canoe.
The resultant velocity is the vector sum of the velocities of the canoe and the river.
Resultant velocity (x-component) = canoe velocity (x-component) + river velocity (x-component)
Resultant velocity (y-component) = canoe velocity (y-component) + river velocity (y-component)
Substituting the given values:
Resultant velocity (x-component) = 0.400 * cos(45°) + 0.450
Resultant velocity (y-component) = 0.400 * sin(45°)
Step 3: Find the magnitude and direction of the resultant velocity.
To find the magnitude of the resultant velocity, use the Pythagorean theorem:
Magnitude of the resultant velocity = sqrt(Resultant velocity (x-component)^2 + Resultant velocity (y-component)^2)
To find the direction of the resultant velocity, use trigonometry:
Direction of the resultant velocity = arctan(Resultant velocity (y-component) / Resultant velocity (x-component))
Substitute the calculated values from steps 2 and 3 to find the final answer.