Get the resultant of the following displacement 6 ft. southwest; 10 ft. north-east; 4 ft. south-east; 12 ft. northwest and 3 ft. southwest. What is the direction of the resultant displacement?
For all of these directions, the x-y displacements are 1/√2 times the distance traveled, so starting at (0,0), you end up at
1/√2 (-6+10+4-12-3 , -6+10+4+12-3) = 1/√2 (-7,17)
The direction, measured from due east, is thus θ, where
tanθ = 17/-7 = 112.3° or N 22.3° W or a heading of 337.7°
Well, that's quite a journey! Let's calculate the resultant displacement for you, but I must warn you, it might take a few laughs along the way.
First, let's break it down:
- 6 ft. southwest: Well, that's simply moving 6 ft. to the south and 6 ft. to the west. So it's like going from California to Texas but making a quick pit stop in New Mexico.
- 10 ft. north-east: This is like taking a stroll from Maine to Florida but getting distracted and ending up in Georgia. A classic detour!
- 4 ft. south-east: Ah, another diagonal move! This means you're going southeast, so it's kind of like trying to blend in with the cool kids in Miami during spring break.
- 12 ft. northwest: Now we've got a good old-fashioned northwest journey. Think about hiking from Washington to Oregon and taking a wrong turn.
- 3 ft. southwest: For this last stretch, we're heading southwest again. Picture yourself bringing some sunscreen to Bora Bora, just to realize you forgot your swimsuit.
Alright, now let's calculate where all these crazy moves lead us.
Adding up all those displacements, we end up with a resultant displacement of approximately 11.48 ft at an angle of 44.03 degrees north of west.
So, to sum it all up, the direction of the resultant displacement is around 44.03 degrees north of west. Just make sure you don't forget your sense of humor when following this wacky path!
To find the resultant of the given displacements, we need to add the individual displacements together and determine the direction of the resultant displacement.
First, let's break down each displacement into its horizontal (east-west) and vertical (north-south) components:
Displacement 1: 6 ft. southwest
Horizontal Component: -6 ft.
Vertical Component: -6 ft.
Displacement 2: 10 ft. northeast
Horizontal Component: 10 ft.
Vertical Component: 10 ft.
Displacement 3: 4 ft. southeast
Horizontal Component: 4 ft.
Vertical Component: -4 ft.
Displacement 4: 12 ft. northwest
Horizontal Component: -12 ft.
Vertical Component: 12 ft.
Displacement 5: 3 ft. southwest
Horizontal Component: -3 ft.
Vertical Component: -3 ft.
Next, add up the horizontal and vertical components:
Horizontal Component: -6 ft. + 10 ft. + 4 ft. - 12 ft. - 3 ft. = -7 ft.
Vertical Component: -6 ft. + 10 ft. - 4 ft. + 12 ft. - 3 ft. = 9 ft.
The resultant horizontal component is -7 ft., and the resultant vertical component is 9 ft.
To find the magnitude (length) of the resultant displacement, we can use the Pythagorean theorem:
Resultant Magnitude = √((-7 ft.)^2 + (9 ft.)^2) = √(49 ft. + 81 ft.) = √130 ft. ≈ 11.40 ft.
The magnitude of the resultant displacement is approximately 11.40 ft.
Finally, to determine the direction of the resultant displacement, we can use trigonometry. The direction can be found by calculating the angle with respect to the positive x-axis (east direction) using the vertical and horizontal components.
Direction = atan(Vertical Component / Horizontal Component)
Direction = atan(9 ft. / -7 ft.)
Direction ≈ -46.34 degrees
The direction of the resultant displacement is approximately -46.34 degrees, measured counterclockwise from the positive x-axis (east direction).
To find the resultant displacement, you need to calculate the net of the individual displacements given.
Let's break down each displacement into its horizontal and vertical components. The positive x-axis represents east, and the positive y-axis represents north. The negative x-axis represents west, and the negative y-axis represents south.
1. 6 ft. southwest: This displacement can be broken down into -6 ft. in the x-axis (west) and -6 ft. in the y-axis (south).
2. 10 ft. north-east: This displacement can be broken down into +10 ft. in the x-axis (east) and +10 ft. in the y-axis (north).
3. 4 ft. south-east: This displacement can be broken down into +4 ft. in the x-axis (east) and -4 ft. in the y-axis (south).
4. 12 ft. northwest: This displacement can be broken down into -12 ft. in the x-axis (west) and +12 ft. in the y-axis (north).
5. 3 ft. southwest: This displacement can be broken down into -3 ft. in the x-axis (west) and -3 ft. in the y-axis (south).
Now, let's sum up the individual horizontal and vertical components separately:
Horizontal component: -6 ft. + 10 ft. + 4 ft. - 12 ft. - 3 ft. = -7 ft.
Vertical component: -6 ft. + 10 ft. - 4 ft. + 12 ft. - 3 ft. = 9 ft.
Now, we have the horizontal and vertical components of the resultant displacement:
Horizontal component: -7 ft.
Vertical component: 9 ft.
To find the magnitude (or length) of the resultant displacement, we can use the Pythagorean theorem:
Resultant displacement magnitude = sqrt((-7 ft.)^2 + (9 ft.)^2)
Resultant displacement magnitude = sqrt(49 ft.^2 + 81 ft.^2)
Resultant displacement magnitude = sqrt(130 ft.^2)
Resultant displacement magnitude ≈ 11.40 ft.
The resultant displacement magnitude is approximately 11.40 ft.
To find the direction of the resultant displacement, you can use trigonometry. The direction can be determined using the tangent function:
Direction (θ) = arctan(vertical component / horizontal component)
Direction (θ) = arctan(9 ft. / -7 ft.)
Using a calculator, the direction (θ) ≈ -51.1°
Therefore, the direction of the resultant displacement is approximately 51.1 degrees south of west.