An airplane is flying due east with a speed of 600 mph. There is a wind blowing at 85 mph at an angle of S 59° E.

Determine the actual speed and angle of the airplane's flight.

To determine the actual speed and angle of the airplane's flight, we need to break this problem down into vector components.

### Step 1: Define the airplane and wind velocity vectors
- **Airplane velocity**: The airplane is flying due east with a speed of 600 mph. This is purely in the east direction, so its velocity vector can be written as:
  \[
  \vec{v}_{\text{airplane}} = (600, 0) \text{ mph}
  \]
  
- **Wind velocity**: The wind is blowing at 85 mph at an angle of S 59° E. This angle means 59° south of the east direction, or 31° below the east axis. We can resolve this wind velocity into its x and y components:
  \[
  \vec{v}_{\text{wind}} = (v_x, v_y)
  \]
  where
  \[
  v_x = 85 \cos(59^\circ)
  \]
  \[
  v_y = -85 \sin(59^\circ)
  \]
  (The y-component is negative because the wind is blowing southward.)

### Step 2: Add the airplane and wind vectors
The total velocity of the airplane will be the vector sum of the airplane's velocity and the wind's velocity:
\[
\vec{v}_{\text{total}} = \vec{v}_{\text{airplane}} + \vec{v}_{\text{wind}} = (600 + v_x, 0 + v_y)
\]

### Step 3: Determine the magnitude and angle of the resultant velocity
The magnitude of the total velocity (i.e., the actual speed of the airplane) is given by:
\[
|\vec{v}_{\text{total}}| = \sqrt{(600 + v_x)^2 + v_y^2}
\]
The direction (angle) of the airplane's flight relative to the east direction is given by:
\[
\theta = \tan^{-1}\left(\frac{v_y}{600 + v_x}\right)
\]

Now, let's calculate these values.import math

# Given values
airplane_speed = 600  # mph
wind_speed = 85       # mph
wind_angle = 59       # degrees south of east

# Resolve wind velocity into components
v_x = wind_speed * math.cos(math.radians(wind_angle))  # east component
v_y = -wind_speed * math.sin(math.radians(wind_angle)) # south component (negative because southward)

# Calculate the resultant velocity components
total_v_x = airplane_speed + v_x
total_v_y = v_y

# Calculate the magnitude (speed) of the resultant velocity
actual_speed = math.sqrt(total_v_x**2 + total_v_y**2)

# Calculate the direction (angle) of the resultant velocity
angle = math.degrees(math.atan2(total_v_y, total_v_x))

# Output the actual speed and angle
actual_speed, angleThe actual speed of the airplane is approximately **647.89 mph**, and the angle of its flight relative to due east is approximately **6.46° south of east**.

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. How would the result change if the wind were blowing at a different angle?
2. What would the speed and direction be if the wind were blowing from the north instead of south?
3. How can we use vector components in 3D motion, such as with airplanes changing altitude?
4. How would the airplane's ground speed change if the wind speed increased?
5. Can you calculate the time it would take for the airplane to travel 1000 miles given its actual speed?

**Tip:** When working with vectors, always break them into components along known axes (like x and y) to simplify complex problems.

Explore how to calculate the actual speed and angle of an airplane's flight when affected by wind velocity. This detailed problem-solving guide includes vector addition, trigonometric calculations, and practical applications. Suitable for high school students and anyone interested in physics and applied mathematics.

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