Vector Addition and Resultant Velocity Techniques

Oct 12, 2024

Lecture Notes on Vector Addition and Resultant Velocity

Understanding Non-Parallel Vectors

  • Vectors that are not parallel cannot have their magnitudes simply added or subtracted.
  • Example illustration provided in lecture.

Methods for Solving Non-Parallel Vector Problems

  1. Scaled Drawing Method

    • Used in the lecture to solve a problem from a past paper (May-June 18, Paper 2-1, Question 1c).
    • Scenario: Finding the resultant velocity of an aircraft in the wind.
    • Details:
      • Aircraft velocity in still air: 95 m/s, heading west.
      • Wind velocity: 28 m/s, 65 degrees south of east.
      • Draw vectors respecting their directions and magnitudes.
    • Steps:
      1. Draw the aircraft vector pointing west.
      2. Draw the wind vector at the specified angle (65 degrees south of east).
      3. Determine the resultant vector by the triangle method.
      4. Use a ruler and protractor to measure lengths and angles.
      5. Resultant velocity found to be around 110 m/s.

  2. Using Solution of Triangles

    • Recommended for more precise calculations compared to scaled drawings.
    • Cosine Rule: Applicable to find the resultant vector using known sides and included angle:
      • Formula: ( b^2 = a^2 + c^2 - 2ac \cdot \cos(B) )
      • Application: Used to compute resultant velocity accurately.
      • Solved example yields resultant velocity approximately 110 m/s.

Key Concepts and Tools

  • Scale Setting: 1 cm = 10 m/s
  • Accuracy Considerations:
    • Importance of minimizing parallax errors.
    • Measuring uncertainty: Error in measurement can lead to a range in the final answer (e.g., ±1 m/s).

Practical Advice

  • Use of protractors and rulers essential for accuracy.
  • Verification using mathematical methods (solution of triangles) helps confirm results.
  • Understanding of scale: Helpful in translating diagram measurements into real-world quantities.

Conclusion

  • Scaled drawing and mathematical verification are both legitimate methods for finding resultant velocities.
  • Method choice depends on the need for precision and available tools.
    • Scaled drawing is straightforward but can be prone to larger uncertainty ranges.
    • Solution of triangles offers precise results, especially useful in exams for verification.