Math Problem Statement
In a 2D game environment an airplane trip involves 3 legs, with two stopovers. The first leg is due east for 620km, the second leg is southeast for 440km and the third leg is at 53° of west, for 550km. What is the plane's total displacement vector by magnitude (km) and angle (°)?
Solution
To find the total displacement vector of the airplane trip, we need to break each leg of the trip into its component vectors and then sum them.
Step 1: Break Down Each Leg into Components
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First Leg (620 km due East)
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Second Leg (440 km Southeast)
- Southeast means south of east.
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Third Leg (550 km at 53° West of North)
- This angle is measured from the north direction, so we convert it to the standard position angle:
Step 2: Sum the Components
Now, sum the x and y components.
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Total x-component:
-
Total y-component:
Step 3: Calculate the Magnitude and Angle of the Resultant Vector
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Magnitude:
-
Angle:
Final Results
- Total Displacement Magnitude: Approximately 481.21 km
- Total Displacement Angle: Approximately 0.46° (measured from the east direction)
If you have any questions or need further details, feel free to ask!
Related Questions
- How would the displacement change if the second leg were instead due south?
- What is the average speed if the trip took 2 hours?
- How would you calculate the total time taken for each leg if you know the speed?
- What would be the displacement if the third leg were directed due west instead?
- How would the total displacement differ in a 3D environment?
Tip: Always break down vector components to simplify calculations of displacement and resultant vectors.
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Math Problem Analysis
Mathematical Concepts
Vector Addition
Trigonometry
Geometry
Formulas
Displacement magnitude: R = √(x_total² + y_total²)
Displacement angle: θ = tan^(-1)(y_total / x_total)
Theorems
Pythagorean Theorem
Vector Resolution Theorem
Suitable Grade Level
Grades 9-12
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