A person walks the path shown below. The total trip consists of four straight-line paths.
Path 1 is 100.0 m due east.
Path 2 is 270.0 m due south.
Path 3 is 140.0 m 30.0° south of west.
Path 4 is 160.0 m 60.0° north of west.
At the end of the walk, what is the person's resultant displacement?
m at ° south of west.
get out some graph paper and draw the line segments.
Or, calculate the x- and y-distances for each part and add them up.
If you end up at (x,y), then tanθ = |y/x|
203.11 m
To find the resultant displacement, we need to add up the horizontal and vertical components of each path and then find the overall displacement.
Path 1: 100.0 m due east.
The horizontal component is 100.0 m to the right, and the vertical component is zero.
Path 2: 270.0 m due south.
The horizontal component is zero, and the vertical component is -270.0 m downward.
Path 3: 140.0 m 30° south of west.
To find the horizontal and vertical components, we need to use trigonometry.
The horizontal component is 140.0 m * cos(30°) = 121.0 m to the left.
The vertical component is 140.0 m * sin(30°) = 70.0 m downward.
Path 4: 160.0 m 60° north of west.
To find the horizontal and vertical components, we need to use trigonometry.
The horizontal component is 160.0 m * cos(60°) = -80.0 m to the left.
The vertical component is 160.0 m * sin(60°) = 138.6 m upward.
Now, let's add up the horizontal and vertical components:
Horizontal component: 100.0 m + 121.0 m + (-80.0 m) = 141.0 m to the right
Vertical component: 0 + (-270.0 m) + (-70.0 m) + 138.6 m = -201.4 m downward
Finally, we can use the horizontal and vertical components to find the magnitude and direction of the resultant displacement:
Magnitude: √(141.0 m)^2 + (-201.4 m)^2 ≈ 243.8 m
Direction: atan((-201.4 m / 141.0 m)) ≈ -52.6°
Therefore, the person's resultant displacement is approximately 243.8 m at 52.6° south of west.
To find the person's resultant displacement, we need to calculate the sum of their displacements in the x-direction (east/west) and the y-direction (north/south).
First, let's break down each path into their x and y components:
Path 1:
Distance = 100.0 m
x-component = 100.0 m (since it is due east, there is no north/south component)
y-component = 0.0 m (as there is no north/south movement)
Path 2:
Distance = 270.0 m
x-component = 0.0 m (since it is due south, there is no east/west component)
y-component = -270.0 m (negative because it is moving south)
Path 3:
Distance = 140.0 m
To find the x and y components, we can use trigonometry.
The angle of the path with respect to the west is 30.0°, and the total distance is 140.0 m.
x-component = 140.0 m * cos(30.0°)
y-component = -140.0 m * sin(30.0°)
Path 4:
Distance = 160.0 m
To find the x and y components, we can again use trigonometry.
The angle of the path with respect to the west is 60.0°, and the total distance is 160.0 m.
x-component = 160.0 m * cos(60.0°)
y-component = 160.0 m * sin(60.0°)
Now, we can add up the x and y components to find the resultant displacement:
Total x-component = 100.0 m + 0.0 m + 140.0 m * cos(30.0°) + 160.0 m * cos(60.0°)
Total y-component = 0.0 m - 270.0 m - 140.0 m * sin(30.0°) + 160.0 m * cos(60.0°)
To find the magnitude and angle of the resultant displacement, we can use the Pythagorean theorem and trigonometry:
Resultant displacement = sqrt((Total x-component)^2 + (Total y-component)^2)
Angle = atan2(Total y-component, Total x-component)
Now you can substitute the values and calculate the resultant displacement and its angle.