To determine the new direction and speed of the plane, we analyze the vectors involved using vector addition. Here's how we solve the problem:

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Step 1: Represent the velocity of the plane

The velocity vector of the plane is given as 250 mph at a bearing of $10^\circ$ east of south. Converting this to Cartesian components:

• The angle relative to the south (negative $y$-axis) is $10^\circ$.
• Breaking the velocity into components: $V_{\text{plane},x} = 250 \sin(10^\circ)$ $V_{\text{plane},y} = -250 \cos(10^\circ)$

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Step 2: Represent the wind velocity

The wind is blowing at 60 mph from the west, so its components are: V_{\text{wind},x} = -60 \quad \text{(west is negative x)} $V_{\text{wind},y} = 0 \quad \text{(no north-south movement)}$

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Step 3: Add the vectors

The resultant velocity components are: $V_{\text{result},x} = V_{\text{plane},x} + V_{\text{wind},x}$ $V_{\text{result},y} = V_{\text{plane},y} + V_{\text{wind},y}$ Substituting: $V_{\text{result},x} = 250 \sin(10^\circ) - 60$ $V_{\text{result},y} = -250 \cos(10^\circ)$

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Step 4: Calculate the magnitude of the resultant velocity

The magnitude of the resultant velocity is: $\|V_{\text{result}}\| = \sqrt{(V_{\text{result},x})^2 + (V_{\text{result},y})^2}$

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Step 5: Calculate the direction

The direction is measured as the angle $\theta$ (relative to the south): $\theta = \tan^{-1}\left(\frac{|V_{\text{result},x}|}{|V_{\text{result},y}|}\right)$ Adjust the angle depending on the quadrant of the resultant vector.

Let’s calculate this step-by-step. math

Plane speed and direction

plane_speed = 250 # mph plane_angle = 10 # degrees (east of south)

Wind speed

wind_speed = 60 # mph

Plane velocity components

V_plane_x = plane_speed * math.sin(math.radians(plane_angle)) # eastward component V_plane_y = -plane_speed * math.cos(math.radians(plane_angle)) # southward component

Wind velocity components

V_wind_x = -wind_speed # westward component V_wind_y = 0 # no north-south component

Resultant velocity components

V_result_x = V_plane_x + V_wind_x V_result_y = V_plane_y + V_wind_y

Magnitude of resultant velocity

V_result_magnitude = math.sqrt(V_result_x2 + V_result_y2)

Direction of resultant velocity (angle relative to south)

theta = math.degrees(math.atan2(V_result_x, -V_result_y)) # angle relative to south

V_result_magnitude, thetaThe new speed and direction of the plane are as follows:

• Speed: $246.76 \, \text{mph}$
• Direction: $3.85^\circ$ west of south (negative indicates westward deviation).

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Detailed Breakdown:

1. The wind reduces the effective eastward speed, causing the plane to drift slightly westward.
2. The new direction deviates slightly from the original $10^\circ$ east of south, ending up at $3.85^\circ$ west of south.

Would you like a detailed step-by-step explanation or visual representation?

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Related Questions:

1. How does wind speed affect the resultant velocity of a moving object?
2. What are the advantages of using vector addition in navigation problems?
3. How would the solution change if the wind blew from the east instead of the west?
4. How can you convert bearings into Cartesian components efficiently?
5. What tools can pilots use to compensate for wind drift in real-time?

Tip: Always double-check the reference direction when interpreting angles in navigation problems.