You have a boat that is capable of moving at 10.4 m/s through still water. You wish to cross a river that flows due west at 2.8 m/s. At what numerical compass heading must you pilot your boat so that you will reach a destination that is due south of your current position?
well, you have to go upstream at an angle of arcSin(2.8/10.4) from the normal. Draw the figure, and think out why this is so.
-2.8 + Vb = -10.4i.
Vb = 2.8 - 10.4i. = 10.8m/s[-74.9o] = 74.9o S. of E. = 15.1o E. of S.
To determine the compass heading you must pilot your boat, we first need to understand the velocity vectors involved.
The boat can move at 10.4 m/s through still water, so we can consider this its maximum speed. Let's call this velocity vector "Vb".
The river flows due west at 2.8 m/s, which means it has a velocity vector pointing westward. Let's call this velocity vector "Vr".
Since we want to reach a destination that is due south, we need to combine the boat's velocity with the river's velocity in such a way that the resultant vector points south.
To find the heading, we need to find the angle between the resultant vector (Vs) and the north direction on the compass. We can use trigonometry to solve this.
1. Subtract the river's velocity vector from the boat's velocity vector to find the resultant vector:
Vs = Vb - Vr
2. Convert the velocity vectors from the Cartesian coordinate system (east, north) to the polar coordinate system (magnitude, direction). This gives us the magnitude and the angle of each vector.
3. Determine the angle between Vs and the north direction on the compass. This angle will give us the compass heading required to reach the destination.
Let's calculate step by step:
Step 1 - Subtract the river's velocity vector from the boat's velocity vector:
Vs = Vb - Vr = (10.4 m/s) - (2.8 m/s west)
Step 2 - Convert the velocity vectors to polar coordinates:
For Vb = 10.4 m/s:
Magnitude: magnitude_b = sqrt((10.4 m/s)^2) = 10.4 m/s
Direction: direction_b = 0 degrees (since it's moving directly north)
For Vr = 2.8 m/s west:
Magnitude: magnitude_r = sqrt((2.8 m/s)^2) = 2.8 m/s
Direction: direction_r = 180 degrees (due west)
Step 3 - Determine the angle between Vs and the north direction:
Since Vs is the resultant vector, we need to find its magnitude and direction:
Magnitude: magnitude_s = sqrt((Vs)^2)
Direction: direction_s = arctan(Vs_y / Vs_x) + 180 degrees
where Vs_x and Vs_y are the x and y components of the resultant vector Vs.
In this case, since the river flows directly west (along the x-axis), Vs_y = magnitude_b * sin(direction_r) and Vs_x = magnitude_b * cos(direction_r).
Substituting the values, we get:
Vs_y = 10.4 m/s * sin(180 degrees) = 0 m/s
Vs_x = 10.4 m/s * cos(180 degrees) = -10.4 m/s (negative because it's in the opposite direction of the boat's movement)
Now, we can calculate the direction of Vs:
direction_s = arctan(0 / -10.4) + 180 degrees
Simplifying, we get:
direction_s = arctan(0) + 180 degrees = 180 degrees
Therefore, the compass heading you must pilot your boat is 180 degrees.