Unit 1.4: Reference Frames and Relative Motion

Introduction

Motion is always described relative to a reference frame. Understanding relative motion helps us analyze situations where different observers experience motion differently. This concept is crucial for solving kinematics problems involving moving objects such as vehicles, boats, and planes.

Key Concepts

Reference Frame: A point of view from which motion is observed and measured.
Relative Velocity: The velocity of an object as seen from another moving reference frame.
Galilean Relativity: In classical physics, all motion is relative unless measured against an absolute reference (which does not exist in classical mechanics).

Tip: When solving problems involving relative motion, carefully define the observer's frame of reference before applying equations.

Relative Velocity

When analyzing moving objects from different reference frames, we use vector addition to determine relative velocity:

If two objects move in the same direction, subtract their speeds.
If two objects move in opposite directions, add their speeds.
For 2D motion, use vector components and vector addition.

The equation for relative velocity is:

\[ v_{\text{A/B}} = v_{\text{A}} - v_{\text{B}} \]

Example:

A person walks at 1.5 m/s inside a train moving at 10 m/s. The person's velocity relative to the ground is (1.5 m/s + 10 m/s) = 11.5 m/s if they walk in the same direction as the train.

Vector Approach to Relative Motion

When dealing with motion at an angle (e.g., boats crossing a river or planes in the wind), we use vector components:

Break each velocity into x and y components.
Use the Pythagorean theorem for the magnitude of the resultant velocity.
Use trigonometry to find direction (angle).

Example:

A boat crosses a river at 5 m/s while the river flows at 3 m/s. The resultant velocity is found using:

\[ v_{\text{resultant}} = \sqrt{(5^2 + 3^2)} \]

The boat's effective speed is 5.83 m/s at an angle \(\theta = \tan^{-1} \left(\frac{3}{5}\right)\).

Mathematical Routines

To calculate relative velocity in two dimensions:

Find x- and y-components of motion.
Apply vector addition using component-wise addition.
Find the resultant speed using Pythagorean theorem.
Determine direction using inverse tangent.

Tip: Draw a vector diagram before attempting calculations.

Scientific Questioning & Argumentation

Critical thinking questions for mastering relative motion:

Conceptual: Why does a person inside a moving car feel stationary?
Application: How does wind affect an airplane’s required heading?
Analysis: If two moving cars measure different speeds for the same event, which one is correct?

Exam Tip: When justifying answers in free-response questions, reference frames of motion and the relative velocity equation.

Practice Activities

Activity 1: Defining Reference Frames

A passenger inside a bus moving at 15 m/s throws a ball forward at 5 m/s. Determine the ball’s velocity relative to:
(A) The passenger
(B) A person standing on the ground

Activity 2: Relative Motion of Two Cars

Car A moves at 30 m/s east, while Car B moves at 20 m/s west. What is the velocity of Car A relative to Car B?

Activity 3: Airplane Navigation

A plane flies at 200 km/h north, while a wind blows at 50 km/h east. Determine the plane’s actual velocity and direction.

Summary & Exam Preparation Tips

In this unit, we explored how motion depends on the observer's reference frame. Key takeaways:

Motion is always described relative to a reference frame.
Relative velocity is found using vector subtraction.
For 2D motion, use vector components and Pythagorean theorem.

Mastering relative motion concepts will help solve airplane, boat, and car motion problems efficiently in AP Physics.