A plane flies 118 km at 15 degrees south of east and then flies 118 km at 35 degrees west of north. Find the magnitude and direction of the total displacement of the plane?
Well, this plane seems to have quite the adventurous spirit, flying in all sorts of directions! Let's calculate its displacement, shall we?
First, let's break down the plane's motion into x and y components.
For the first leg of the journey, flying 118 km at 15 degrees south of east, the x-component will be 118 km * cos(15°) and the y-component will be -118 km * sin(15°).
For the second leg, flying 118 km at 35 degrees west of north, the x-component will be -118 km * sin(35°) and the y-component will be 118 km * cos(35°).
Now, let's add up the x and y components to find the total displacement of the plane.
The total x-component is (118 km * cos(15°)) + (-118 km * sin(35°)) = -51.71 km.
The total y-component is (-118 km * sin(15°)) + (118 km * cos(35°)) = 4.57 km.
To find the magnitude of the total displacement, we can use the Pythagorean theorem:
Magnitude = sqrt((-51.71 km)^2 + (4.57 km)^2) = 51.99 km (rounded to the nearest hundredth).
To find the direction of the total displacement, we can use inverse tangent:
Direction = atan(4.57 km / -51.71 km) = -5.10 degrees (rounded to the nearest tenth).
So, the magnitude of the total displacement is approximately 51.99 km, and the direction is approximately -5.10 degrees. And here's a joke for you: Why don't airplanes get bored? Because they always have wings to pass the time!
To find the magnitude and direction of the total displacement of the plane, we can break down the two legs of the plane's flight into their x and y components.
First, let's find the x and y components of the first leg of the flight, which is 118 km at 15 degrees south of east.
The x component can be found using the cosine function:
x = 118 km * cos(15 degrees)
x ≈ 112.885 km
The y component can be found using the sine function:
y = 118 km * sin(15 degrees)
y ≈ -30.622 km
Similarly, let's find the x and y components of the second leg of the flight, which is 118 km at 35 degrees west of north.
The x component can be found using the sine function (since it is west of north):
x = 118 km * sin(35 degrees)
x ≈ 67.793 km
The y component can be found using the cosine function (since it is west of north):
y = 118 km * cos(35 degrees)
y ≈ 95.033 km
Now, let's add up the x and y components to find the total displacement:
Total x component = 112.885 km - 67.793 km
Total x component ≈ 45.092 km
Total y component = -30.622 km + 95.033 km
Total y component ≈ 64.411 km
To find the magnitude of the total displacement, we can use the Pythagorean theorem:
Magnitude = sqrt((Total x component)^2 + (Total y component)^2)
Magnitude ≈ sqrt((45.092 km)^2 + (64.411 km)^2)
Magnitude ≈ sqrt(2035.793 km^2)
Magnitude ≈ 45.13 km (rounded to two decimal places)
To find the direction of the total displacement, we can use the inverse tangent function:
Direction = atan(Total y component / Total x component)
Direction ≈ atan(64.411 km / 45.092 km)
Direction ≈ atan(1.428)
Direction ≈ 54.13 degrees (rounded to two decimal places)
Therefore, the magnitude of the total displacement of the plane is approximately 45.13 km, and the direction is approximately 54.13 degrees.
To find the magnitude and direction of the total displacement of the plane, we can break down the two legs of the journey into their x and y components.
First, let's analyze the first leg of the journey: flying 118 km at 15 degrees south of east.
Since the plane is traveling south of east, we need to calculate the x and y components separately.
The x-component can be calculated using the cosine of the angle:
x₁ = 118 km * cos(15°) ≈ 113.81 km
The y-component can be calculated using the sine of the angle:
y₁ = 118 km * sin(15°) ≈ 30.48 km
Next, let's analyze the second leg of the journey: flying 118 km at 35 degrees west of north.
Similarly, we need to calculate the x and y components separately.
Since the plane is traveling west of north, we will need to use negative values for the x-component.
The x-component can be calculated using the cosine of the angle:
x₂ = -118 km * cos(35°) ≈ -96.34 km
The y-component can be calculated using the sine of the angle:
y₂ = 118 km * sin(35°) ≈ 67.78 km
Now we can find the total x and y components by adding the respective components of the two legs:
x_total = x₁ + x₂ ≈ 113.81 km - 96.34 km ≈ 17.47 km
y_total = y₁ + y₂ ≈ 30.48 km + 67.78 km ≈ 98.26 km
To find the magnitude of the total displacement, we can use the Pythagorean theorem:
magnitude = √(x_total² + y_total²) ≈ √(17.47 km)² + (98.26 km)² ≈ 99.49 km
To find the direction of the total displacement, we can use the inverse tangent function:
direction = tan⁻¹(y_total / x_total) ≈ tan⁻¹(98.26 km / 17.47 km) ≈ 79.36°
Therefore, the magnitude of the total displacement of the plane is approximately 99.49 km, and the direction is approximately 79.36°.
All angles are measured CCW from +x-axis.
D = 118km[345o] + 118[125o] =
114-30.5i + -67.7+96.7i = 46.3 + 66.2i = 80.8km[55o].