an air plane takes a circuitous route. the journey is split into 4 sections.
d1= 6000km (30.0 n of e)
d2= 5600km (NW)
d3= 4500km (35.0 S of W)
d4= 7000km (12.0 W of S)
- Find the resultant displacement of the air plane
what do i do, and what is the coreect answer for the planes resultant?
You add the vectors. I recommend break each of the distances into N and E vectors, so adding the four vectors can be done in components. I will be happy to critique your thinking or work.
To find the resultant displacement of the airplane, we need to add up the individual displacements from each section.
First, let's convert the given directions into a common coordinate system using the compass rose:
- d1: 30.0 degrees north of east
- d2: northwest (which can be expressed as 315.0 degrees)
- d3: 35.0 degrees south of west
- d4: 12.0 degrees west of south
Now, let's break down each displacement into its horizontal (x) and vertical (y) components. We'll use trigonometry to do this.
For d1:
- x1 = cos(30.0) * 6000 km (since cos(theta) = adjacent/hypotenuse)
- y1 = sin(30.0) * 6000 km (since sin(theta) = opposite/hypotenuse)
For d2:
- x2 = cos(315.0) * 5600 km
- y2 = sin(315.0) * 5600 km
For d3:
- x3 = cos(35.0) * 4500 km
- y3 = sin(35.0) * 4500 km
For d4:
- x4 = cos(192.0) * 7000 km (as 12.0 degrees is 180.0 + 12.0 = 192.0 degrees)
- y4 = sin(192.0) * 7000 km
Now, we can sum up the x and y components separately:
- X = x1 + x2 + x3 + x4
- Y = y1 + y2 + y3 + y4
Finally, we can calculate the resultant displacement (R) using the Pythagorean theorem:
- R = √(X^2 + Y^2)
Now, let's find the values:
x1 = cos(30.0) * 6000 = 5196.152 km
y1 = sin(30.0) * 6000 = 3000 km
x2 = cos(315.0) * 5600 = -3964.935 km
y2 = sin(315.0) * 5600 = -3964.935 km
x3 = cos(35.0) * 4500 = -3660.269 km
y3 = sin(35.0) * 4500 = -2577.816 km
x4 = cos(192.0) * 7000 = -6531.683 km
y4 = sin(192.0) * 7000 = -2316.569 km
X = 5196.152 - 3964.935 - 3660.269 - 6531.683 = -8960.735 km
Y = 3000 - 3964.935 - 2577.816 - 2316.569 = -5869.32 km
R = √((-8960.735)^2 + (-5869.32)^2)
R = √(80237155.91 + 34393101.57)
R = √114430257.48
R ≈ 10,699.123 km
Therefore, the resultant displacement of the airplane is approximately 10,699.123 km.