An helicopter flies in a strange pattern, all at the same altitude of 5 km: 1) it flies 29 km in a direction 30.0° east of north, 2) it flies 20 km due south, and 3) it flies 10 km in a direction 30.0° north of west. How far did the helicopter end up from its starting point?
All angles are measured CCW from due east.
D = 29km[60o] + 20km[270o] + 10km[150o] =
To find the distance the helicopter ended up from its starting point, we can break down the helicopter's motion into components along the north-south (y-axis) and east-west (x-axis) directions.
Let's analyze each leg of the helicopter's flight:
1) The helicopter flies 29 km in a direction 30.0° east of north:
- The distance along the y-axis (north-south) can be calculated as: 29 km * sin(30°) = 14.5 km
- The distance along the x-axis (east-west) can be calculated as: 29 km * cos(30°) = 25.1 km
2) The helicopter flies 20 km due south:
- The distance along the y-axis (north-south) is -20 km since it goes in the opposite (south) direction.
- There is no distance along the x-axis (east-west) since the helicopter doesn't move in that direction.
3) The helicopter flies 10 km in a direction 30.0° north of west:
- The distance along the y-axis (north-south) can be calculated as: 10 km * sin(30°) = 5 km
- The distance along the x-axis (east-west) can be calculated as: 10 km * cos(30°) = 8.7 km
Now, let's add up the displacements along the x-axis and y-axis separately to find the net displacement:
Net displacement along the y-axis (north-south):
= 14.5 km - 20 km + 5 km
= -0.5 km (since it ended up slightly south)
Net displacement along the x-axis (east-west):
= 25.1 km + 0 km + 8.7 km
= 33.8 km (since it ended up slightly east)
Finally, we can use the Pythagorean theorem to find the magnitude of the net displacement:
Distance = sqrt((Net displacement along the x-axis)^2 + (Net displacement along the y-axis)^2)
Distance = sqrt((33.8 km)^2 + (-0.5 km)^2)
Distance = sqrt(1140.44 km^2 + 0.25 km^2)
Distance = sqrt(1140.69 km^2)
Distance ≈ 33.8 km
Therefore, the helicopter ended up approximately 33.8 km from its starting point.