A woman sailing a boat at night is following directions to a dock. The instructions read to first sail 28.0 m in a direction 62 degrees north of east from her current location, and then travel 25.0m in a direction 110 degrees north of east.
If the woman makes a mistake and travels in the opposite direction for the second leg of the trip, where will she end up relative to her starting position measured clockwise from the east direction?
R = ?
Theta = ?
Compass direction = ?
To determine the woman's final position relative to her starting position, we need to calculate the resultant displacement vector. This can be done by adding the individual displacement vectors together.
First, let's convert the given directions into components using trigonometry.
For the first leg of the trip:
Magnitude = 28.0 m
Angle = 62 degrees north of east
To calculate the x-component:
x1 = Magnitude * cos(Angle)
x1 = 28.0 m * cos(62 degrees)
x1 ≈ 13.218 m
To calculate the y-component:
y1 = Magnitude * sin(Angle)
y1 = 28.0 m * sin(62 degrees)
y1 ≈ 23.992 m
So, the first displacement vector can be represented as (x1, y1) = (13.218 m, 23.992 m).
For the second leg of the trip:
Magnitude = 25.0 m
Angle = 110 degrees north of east
To calculate the x-component:
x2 = Magnitude * cos(Angle)
x2 = 25.0 m * cos(110 degrees)
x2 ≈ -9.369 m
To calculate the y-component:
y2 = Magnitude * sin(Angle)
y2 = 25.0 m * sin(110 degrees)
y2 ≈ 21.464 m
Since the woman made a mistake and traveled in the opposite direction for the second leg, the sign of the x-component should be negative.
So, the second displacement vector can be represented as (-x2, y2) = (-(-9.369 m), 21.464 m) = (9.369 m, 21.464 m).
To calculate the resultant displacement vector, we add the individual displacement vectors together:
R = (x1 + x2, y1 + y2)
R = (13.218 m + 9.369 m, 23.992 m + 21.464 m)
R ≈ (22.587 m, 45.456 m)
To find the magnitude and direction of the resultant displacement, we can use the Pythagorean theorem and trigonometry.
Magnitude:
Magnitude (R) = sqrt((x1 + x2)^2 + (y1 + y2)^2)
Magnitude (R) = sqrt((22.587 m)^2 + (45.456 m)^2)
Magnitude (R) ≈ 50.654 m
Direction:
Theta = atan2((y1 + y2), (x1 + x2))
Theta = atan2((23.992 m + 21.464 m), (13.218 m + 9.369 m))
Theta ≈ 59.822 degrees
Since we are measuring the direction clockwise from the east direction, the compass direction can be determined by subtracting the calculated angle from 90 degrees.
Compass direction = 90 degrees - Theta
Compass direction ≈ 30.178 degrees
Therefore, if the woman makes a mistake and travels in the opposite direction for the second leg of the trip, she will end up approximately 50.654 m away from her starting position, with a direction of approximately 30.178 degrees clockwise from the east direction.