PHY101 Mechanics | Recitation 4: Pursuit Problems, Vector Algebra & Coordinate Rotations

Описание: In this recitation session for PHY101: Mechanics, we move from theory to application by solving rigorous problems in kinematics and vector space. The session begins with a "Policeman vs. Speeder" pursuit problem, emphasizing the importance of time delays (t_d) and initial conditions in motion equations.

The second half of the lecture focuses on Vector Manipulation. The instructor provides a masterclass on describing the same vector in different coordinate systems. You will learn the mathematical relationship between components in a standard frame (x, y) versus a rotated frame (x', y'). This is a foundational concept for understanding how physical laws remain "invariant" under coordinate transformations.

Key Learning Outcomes:

◆ Solve pursuit problems involving constant velocity vs. uniform acceleration with time offsets.

◆ Learn why standard kinematic equations like x = ut + ½at² fail if the object doesn't start at t = 0.

◆ Master the use of Unit Vectors (ex, ey, ez) as building blocks in 3D space.

◆ Calculate vector magnitudes using the 3D Pythagorean Theorem.

◆ Derive the Rotation Matrix logic: Relating Ax, Ay to Ax', Ay' via rotation angle θ.

Lecture Timestamps:

[00:10] Kinematics Case Study: The Policeman and the Speeder.

[01:44] Visualizing Motion: Plotting Velocity-Time (v-t) graphs for pursuit.

[04:41] Calculating separation distances using the Area Under the Curve.

[08:58] The "Time Offset" Trap: Why lower limits of integration matter.

[14:52] Using Algebra to find the "Catch-up Time" (T).

[17:41] 3D Vector Operations: Components, unit vectors, and magnitude.

[23:55] Resultant Vectors and solving for unknown components (Bz).

[33:02] Advanced Topic: Rotating Coordinate Axes (The x'-y' system).

[41:40] Expressing ex' and ey' in terms of ex and ey.

[48:51] Checking results with boundary cases (θ = 0° and θ = 90°).