A chartered airplane flies 100 km east, then 350 km northwest, and then 150 km at 30°
west of south. After an additional displacement, the plane is back where it started. Find the
magnitude and the direction of the last leg of the journey.
To find the magnitude and direction of the last leg of the journey, we need to consider the combined displacement of the chartered airplane.
Let's break down the given information:
1. The airplane flies 100 km east.
2. The airplane then flies 350 km northwest. This implies that the airplane travels 350 km at an angle of 45 degrees to the west-northwest direction.
3. The airplane then flies 150 km at 30 degrees west of south.
To visualize the displacements, we can draw a vector diagram:
---------------- | 45° NW
| |
| |
| |
| |
| |
| |
|-------------------------
Now, to find the magnitude and direction of the last leg, we need to find the resultant displacement vector by adding the given vectors.
1. Convert the 45° northwest vector into its horizontal and vertical components.
Since it's at 45°, both horizontal and vertical components will have the same value (350 km).
The horizontal component (west) will be 350 * cos(45°) and the vertical component (north) will be 350 * sin(45°).
Horizontal component = 350 km * cos(45°) = 350 km * 0.707 ≈ 247.5 km (rounded to one decimal place)
Vertical component = 350 km * sin(45°) = 350 km * 0.707 ≈ 247.5 km (rounded to one decimal place)
2. Convert the 150 km at 30° west of south vector into its horizontal and vertical components.
To determine the components, we need to consider that it is "west of south." That means it's in the fourth quadrant.
The horizontal component (west) will be 150 km * cos(30°) since it is west of the south direction.
The vertical component (south) will be 150 km * sin(30°) since it is in the south direction.
Horizontal component = 150 km * cos(30°) ≈ 130 km (rounded to the nearest whole number)
Vertical component = 150 km * sin(30°) = 150 km * 0.5 = 75 km
3. Add the horizontal and vertical components of all vectors to find the final displacement.
Horizontal component: 100 km (east) + 247.5 km (west from the northwest vector) + 130 km (west from the west of south vector) = 477.5 km (rounded to one decimal place)
Vertical component: 0 km (north) + 247.5 (north from the northwest vector) - 75 km (south from the west of south vector) = 172.5 km (rounded to one decimal place)
Now we can find the magnitude and direction of the resultant displacement using the Pythagorean theorem and trigonometry.
Magnitude (resultant displacement) = sqrt((477.5 km)^2 + (172.5 km)^2) ≈ 514.7 km (rounded to one decimal place)
Direction (angle) = arctan(172.5 km / 477.5 km) ≈ 20.6° (rounded to one decimal place)
Therefore, the magnitude of the last leg of the journey is approximately 514.7 km, and the direction is approximately 20.6°.