A person walks the path shown below. The total trip consists of four straight-line paths.
Path 1 is 80.0 m due east.
Path 2 is 250.0 m due south.
Path 3 is 130.0 m 30.0° south of west.
Path 4 is 190.0 m 60.0° north of west.
At the end of the walk, what is the person's resultant displacement?
Well, it seems like this person went on quite the adventure! To find the resultant displacement, we need to combine all the individual displacements.
Let's break it down path by path:
Path 1: 80.0 m due east
Path 2: 250.0 m due south
Path 3: 130.0 m 30.0° south of west
Path 4: 190.0 m 60.0° north of west
Now, let's do some math magic to find the resultant displacement.
Path 1 (80.0 m due east) is simply 80.0 m east.
Path 2 (250.0 m due south) is simply 250.0 m south.
Path 3 (130.0 m 30.0° south of west) requires some trigonometry. The south component can be found using sin(30°) = opposite/hypotenuse, which gives us opposite = sin(30°) x 130.0 m.
So, the south component of Path 3 is (sin(30°) x 130.0) m.
Finally, for Path 4 (190.0 m, 60.0° north of west), we need to find the north component. We can use sin(60°) = opposite/hypotenuse, which gives us opposite = sin(60°) x 190.0 m.
So, the north component of Path 4 is (sin(60°) x 190.0) m.
Now, add up all the east and west displacements to get the east-west component:
East component = Path 1 + Path 3 = 80.0 m + 0 m (since it is south of west) = 80.0 m east in total.
Add up all the north and south displacements to get the north-south component:
South component = Path 2 + Path 4 = 250.0 m + 0 m (since it is north of west) = 250.0 m south in total.
Finally, let's calculate the resultant displacement, which is the hypotenuse of the right triangle formed by the east-west and north-south components.
Using the Pythagorean theorem, we have:
Resultant displacement = √(East component² + South component²) = √(80.0 m² + 250.0 m²) ≈ 264.94 m
So, after all that walking, the person's resultant displacement is approximately 264.94 meters. That's quite a journey!
To find the resultant displacement, we need to add up the individual displacements vectorially. Let's break down each displacement into its components.
Path 1:
- Displacement in the x-direction (east): 80.0 m
- Displacement in the y-direction (north): 0 m (no displacement)
Path 2:
- Displacement in the x-direction (east): 0 m (no displacement)
- Displacement in the y-direction (north): -250.0 m (south)
Path 3:
- Displacement in the x-direction (east): -130.0 m*cos(30°) = -112.6 m (westward component)
- Displacement in the y-direction (north): -130.0 m*sin(30°) = -65.0 m (southward component)
Path 4:
- Displacement in the x-direction (east): -190.0 m*cos(60°) = -95.0 m (westward component)
- Displacement in the y-direction (north): 190.0 m*sin(60°) = 164.9 m (northward component)
Now, we can add up the x and y components separately:
Displacement in the x-direction: 80.0 m - 112.6 m - 95.0 m = -127.6 m (west)
Displacement in the y-direction: 0 m - 250.0 m - 65.0 m + 164.9 m = -150.1 m (south)
Now, we can find the magnitude and direction of the resultant displacement:
Magnitude of the resultant displacement = sqrt((-127.6 m)^2 + (-150.1 m)^2) = 197.8 m
Direction of the resultant displacement = arctan((-150.1 m)/(-127.6 m)) = 49.3° south of west
Therefore, the person's resultant displacement is 197.8 m 49.3° south of west.
To find the person's resultant displacement, we need to add up the individual displacements of the four paths. The resultant displacement can be determined by breaking each path into its horizontal (east/west) and vertical (north/south) components.
Let's start by converting paths 3 and 4 into their horizontal and vertical components.
Path 3:
The magnitude of path 3 is 130.0 m, and it is at an angle of 30.0° south of west.
To find the horizontal component, we need to find the cosine of the angle:
horizontal component = 130.0 m * cos(30.0°)
horizontal component = 130.0 m * 0.866
horizontal component = 112.58 m (rounded to two decimal places)
To find the vertical component, we need to find the sine of the angle:
vertical component = 130.0 m * sin(30.0°)
vertical component = 130.0 m * 0.5
vertical component = 65.00 m
Path 4:
The magnitude of path 4 is 190.0 m, and it is at an angle of 60.0° north of west.
To find the horizontal component, we need to find the cosine of the angle:
horizontal component = 190.0 m * cos(60.0°)
horizontal component = 190.0 m * 0.5
horizontal component = 95.00 m
To find the vertical component, we need to find the sine of the angle:
vertical component = 190.0 m * sin(60.0°)
vertical component = 190.0 m * 0.866
vertical component = 164.14 m (rounded to two decimal places)
Now, let's add up the horizontal and vertical components of all paths to find the resultant displacement:
Horizontal component = Path 1 + Path 3 (horizontal) + Path 4 (horizontal)
= 80.0 m + 112.58 m + 95.00 m
= 287.58 m
Vertical component = Path 2 + Path 3 (vertical) + Path 4 (vertical)
= -250.0 m + 65.00 m + 164.14 m
= -20.86 m (rounded to two decimal places)
Finally, we can find the magnitude and direction (angle) of the resultant displacement using the horizontal and vertical components:
Resultant displacement magnitude = √(horizontal component^2 + vertical component^2)
= √(287.58 m^2 + (-20.86 m)^2)
≈ 288.11 m (rounded to two decimal places)
To find the direction, we can use the arctan function:
Direction = arctan(vertical component / horizontal component)
= arctan((-20.86 m) / 287.58 m)
≈ -4.13° (rounded to two decimal places)
Therefore, the person's resultant displacement is approximately 288.11 m at an angle of -4.13° with respect to east (clockwise from the positive x-axis).
Start the person at (0,0), and calculate the new position after each path.
After path 1: (80,0)
After path 2: (80, -250)
After path 3: x displacement is -130*cos30; y displacement is -130*sin30
(80 - 130*cos30, -250 -130*sin30)
After path 4: x displacement is -190*cos30, y displacement is 190*sin30;
New position is:
(80 - 130*cos30 - 190*cos30, -250 -130*sin30 + 190*sin30)
The magnitude of the displacement is (x^2 + y^2)^0.5, where (x,y) is the final position