Vector Motion of a Plane

Introduction

In this lesson, we will delve into a practical example involving vector motion, displacement, average velocity, and average speed.

Scenario Description

A plane's journey involves two segments:

1. First Segment: Traveling 483 kilometers east from City A to City B in 45 minutes.
2. Second Segment: Flying 966 kilometers south from City B to City C in one and a half hours.

We will define east as aligned with the unit vector i and north with the unit vector j.

Displacement Calculation (Vector Notation)

Part A: Magnitude and Direction of Plane's Displacement

• • Vector Notation:First segment: 483 i kilometers.
  • Second segment: -966 j kilometers.
• Total Displacement (R): R = 483 i - 966 j kilometers.
• Magnitude: |R| = sqrt(483² + (-966)²) kilometers ≈ 1080 kilometers.
• Direction: θ = tan^(-1)(-966/483) ≈ -63.4 degrees from the positive x-axis.

Average Velocity Calculation

Part B: Magnitude and Direction of Average Velocity

• Formula: Average Velocity = Displacement / Total Time.
• Calculation: V = (483 i - 966 j) / 2.25 hours = 215 i - 429 j kilometers per hour.
• Magnitude: |V| = sqrt(215² + (-429)²) kilometers per hour ≈ 480 kilometers per hour.
• Direction: Same as displacement, -63.4 degrees from the positive x-axis.

Average Speed Calculation

Part C: Average Speed

• Formula: Average Speed = Total Distance / Total Time.
• Calculation: (483 km + 966 km) / 2.25 hours ≈ 644 kilometers per hour.

Conclusion

This example illustrates the application of vector notation in solving problems related to displacement, velocity, and speed in physics, an essential aspect of studying motion and kinematics.