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Question

A plane is traveling in the direction of 35⁰ north of west at a speed of 225 mph. A wind is blowing in the direction of 20⁰ south of west at 40 mph. Find the plane’s new direction and speed. Show all work.

(please include detailed diagrams and explanations!!)

Answer 1

We can solve this by using the vector addition, breaking down each velocity into east-west (x) and north-south (y) components.

Step 1: Set up a coordinate system

1. East = positive x
2. North = positive y
3. West and South are negative

Step 2: Resolve the plane's velocity.

1. Plane speed = 225 mph
2. Direction = 35° north of west. So that means that:
3. x_p = -225 cos(35°)
4. y_p = +225 sin(35°)
5. Numerically, it would be:
6. x_p ≈ -225 (0.8192) = -184.32
7. y_p ≈ 225 (0.5736) = 129.06

Step 3: Resolve the wind's velocity

1. Wind speed = 40 mph
2. Direction = 20° south of west
3. x_w = -40 cos(20°)
4. y_w = -40 sin(20°)
5. Numerically:
6. x_w ≈ -40 (0.9397) = -37.59
7. y_w ≈ +40 (0.3420) = -13.68

Step 4: Adding all the vectors together

1. x_total = -184.32 - 37.59 = -221.91
2. y_total = 129.06 - 13.68 = 115.38

Step 5: Finding the new speed, which is

1. v = √(-221.91)²+(115.38)²
2. v ≈ √49244 + 13312
3. v ≈ √62556 ≈ 250.1 mph

Step 6: Finding the new direction

1. The angle relative to the west would be:
2. θ = tan^-1 (115.38 / 221.91)
3. θ ≈ 27.5°
4. Since the y component is positive, the direction is north of west.

So that means that the final answer should be approximately 250 mph for the new speed, and approximately 27.5º north of west towards the new direction.