On an unwanted weather situation, rescue operators fly a helicopter that is aimed to travel 100 knots due southwest. However, two winds coming from different directions move it sideways. One wind is making it move sideways 20 knots northward while the other makes it move 10 knots, 600 south of west. In what speed and direction will the helicopter fly?
Vr = 100[225o] + 20[90o] + 10[240o].
Vr = -70.71-70.71i + 20i -5-8.66i,
Vr = -75.71 - 79.37i = 110knots[46.35o] S. of W.
Correction:
Vr = -75.71 - 59.37i = 96knots[38.1o] S. of W.
Well okay
To determine the speed and direction at which the helicopter will fly, we need to break down the motion caused by the wind into its horizontal and vertical components. Let's analyze the situation step by step:
1. Start by considering the desired motion of the helicopter. It aims to travel at a speed of 100 knots due southwest. Southwest refers to a direction that is 45 degrees south of due west.
2. Determine the horizontal and vertical components of the motion caused by each wind:
- Wind 1: This wind is making the helicopter move sideways 20 knots northward. Since it's only affecting the vertical direction, we don't need to consider its horizontal component here.
- Wind 2: This wind makes the helicopter move 10 knots, 600 south of west. To determine its horizontal and vertical components, we need to break down the motion using trigonometry. The angle between the west direction and the motion is given by the inverse tangent of the ratio of the two sides of the right triangle formed (south and west).
Angle = arctan(600 / 10) = arctan(60) ≈ 60.95 degrees.
Now we can determine the horizontal and vertical components:
Horizontal component = 10 knots * cos(60.95 degrees) ≈ -5 knots (westward)
Vertical component = 10 knots * sin(60.95 degrees) ≈ -8.66 knots (southward)
3. Combine the horizontal and vertical components to find the net effect of the winds on the helicopter's motion:
- Horizontal component: The desired southwest motion is 45 degrees south of due west. Wind 2, with a horizontal component of -5 knots, affects this motion. Therefore, the net horizontal component is 45 degrees - (-5 degrees) = 50 degrees south of due west.
- Vertical component: Wind 1 causes the helicopter to move 20 knots northward, while Wind 2 causes it to move 8.66 knots southward. So the net vertical component is 20 - 8.66 = 11.34 knots northward.
4. Finally, determine the magnitude and direction of the resulting motion:
The magnitude of the motion is given by the Pythagorean theorem:
Magnitude = sqrt((50 knots)^2 + (11.34 knots)^2) ≈ 51.97 knots
The direction is given by the inverse tangent of the horizontal and vertical components:
Direction = arctan(11.34 knots / 50 knots) ≈ 12.95 degrees north of due west.
Therefore, the helicopter will fly at a speed of approximately 51.97 knots in a direction approximately 12.95 degrees north of due west.