A person walks the path shown below. The total trip consists of four straight-line paths.
Path 1 is 30.0 m due east.
Path 2 is 200.0 m due south.
Path 3 is 140.0 m 30.0° south of west.
Path 4 is 170.0 m 60.0° north of west.
At the end of the walk, what is the person's resultant displacement?
m at ° south of west.
To find the person's resultant displacement, we need to calculate the horizontal and vertical components of each path and then add them up.
Let's start by representing the given paths using a coordinate system, where east is positive x and north is positive y.
Path 1: 30.0 m due east
The horizontal component is 30.0 m in the positive x direction (east), and the vertical component is 0 m.
Path 2: 200.0 m due south
The horizontal component is 0 m, and the vertical component is -200.0 m (negative y direction).
Path 3: 140.0 m 30.0° south of west
To find the horizontal and vertical components, we need to convert the given angle to its x and y components. The given angle is south of west, which means it's in the third quadrant. The angle from the positive x-axis to the path is (-30.0°) + 180° = 150.0°.
The horizontal component is 140.0 m * cos(150.0°), which equals -122.0 m (negative x direction).
The vertical component is 140.0 m * sin(150.0°), which equals -70.0 m (negative y direction).
Path 4: 170.0 m 60.0° north of west
Similar to Path 3, we need to convert the given angle to its x and y components. The angle from the positive x-axis to the path is (-60.0°) + 180° = 120.0°.
The horizontal component is 170.0 m * cos(120.0°), which equals -85.0 m (negative x direction).
The vertical component is 170.0 m * sin(120.0°), which equals 147.4 m (positive y direction).
Now, we can add up the horizontal and vertical components to find the resultant displacement:
Horizontal component = 30.0 m + (-122.0 m) + (-85.0 m) = -177.0 m
Vertical component = 0 m + (-200.0 m) + (-70.0 m) + 147.4 m = -122.6 m
To find the magnitude of the resultant displacement, we can use the Pythagorean theorem:
Resultant displacement = √(horizontal component^2 + vertical component^2)
Resultant displacement = √((-177.0 m)^2 + (-122.6 m)^2)
Resultant displacement ≈ 214.5 m
To find the direction of the resultant displacement, we can use trigonometry:
Direction = atan(vertical component / horizontal component)
Direction = atan(-122.6 m / -177.0 m)
Direction ≈ 35.3° south of west
Therefore, the person's resultant displacement is approximately 214.5 m at 35.3° south of west.