Questions: What is the total displacement for a boat that sails 200.0m [S25 W] and then tacks and sails 150.0m [30° E]?

What is the total displacement for a boat that sails 200.0m [S25 W] and then tacks and sails 150.0m [30° E]?
Transcript text: What is the total displacement for a boat that sails 200.0m [S25 W] and then tacks and sails 150.0m [30° E]?
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Solution

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Solution Steps

Step 1: Break Down the Vectors into Components

  • For the first vector, 200.0 m [S 25° W]:

    • South component: \( 200.0 \times \cos(25^\circ) \)
    • West component: \( 200.0 \times \sin(25^\circ) \)

  • For the second vector, 150.0 m [N 30° E]:

    • North component: \( 150.0 \times \cos(30^\circ) \)
    • East component: \( 150.0 \times \sin(30^\circ) \)
Step 2: Calculate the Net Components

  • Net North-South component:

    • Southward: \( 200.0 \times \cos(25^\circ) \)
    • Northward: \( 150.0 \times \cos(30^\circ) \)
    • Net: \( 150.0 \times \cos(30^\circ) - 200.0 \times \cos(25^\circ) \)

  • Net East-West component:

    • Westward: \( 200.0 \times \sin(25^\circ) \)
    • Eastward: \( 150.0 \times \sin(30^\circ) \)
    • Net: \( 150.0 \times \sin(30^\circ) - 200.0 \times \sin(25^\circ) \)
Step 3: Calculate the Magnitude of the Resultant Displacement
  • Use the Pythagorean theorem to find the magnitude of the resultant vector:
    • \( \text{Magnitude} = \sqrt{(\text{Net North-South})^2 + (\text{Net East-West})^2} \)

Final Answer

\(\boxed{75.0 \, \text{m}} \)

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