Relative motion basics

Relative motion describes how the motion of one object appears when observed from another moving object, called the reference frame. In physics, all motion is relative—there is no absolute reference point in the universe.

Key Definitions:

• Reference Frame: The viewpoint or coordinate system from which motion is observed and measured
• Relative Velocity: The velocity of object A as observed from object B's reference frame
• Observer: The person or object from whose perspective motion is being analyzed

Fundamental Equation:

v⃗_AB = v⃗_A - v⃗_B_

Where:

• v⃗_AB = velocity of A relative to B
• v⃗_A = velocity of A relative to ground (absolute velocity)
• v⃗_B = velocity of B relative to ground (absolute velocity)_

Vector Nature: Relative velocity is a vector quantity, meaning direction matters critically. When objects move in the same direction, you subtract speeds; when moving in opposite directions, you add speeds.

Important Principle: The velocity of B relative to A is the negative of A relative to B: v⃗_BA = -v⃗_AB

Memory Aid - ROAR: Reference frame, Observer perspective, Always vectors, Remember subtraction

2D Relative Motion: When objects move at angles, use vector addition with components (i and j notation) or the Pythagorean theorem and trigonometry to find magnitude and direction of relative velocity.

Key Insight: An object stationary in one reference frame appears moving in another frame—this concept underlies Einstein's relativity theories.

Walking on a Moving Train 🚂

Imagine walking forward at 2 m/s inside a train moving at 30 m/s. To someone on the platform, you're moving at 32 m/s forward. But to someone sitting on the train, you're only moving at 2 m/s. Your motion depends entirely on the observer's reference frame.

River Crossing Problem 🛶

A swimmer crossing a river faces relative motion in two dimensions. If the river flows east at 3 m/s and the swimmer swims north at 4 m/s (relative to water), their actual velocity relative to the riverbank is 5 m/s at 53° from east (using Pythagorean theorem: √(3² + 4²) = 5). The swimmer aims one direction but actually travels another!

Aircraft Navigation ✈️

Pilots constantly deal with relative motion. An aircraft's airspeed (speed relative to air) differs from groundspeed (speed relative to ground) due to wind. A plane flying north at 200 m/s with a 50 m/s eastward wind has a groundspeed of approximately 206 m/s at an angle from north.

Overtaking Cars 🚗

Two cars traveling at 25 m/s and 30 m/s in the same direction have a relative velocity of only 5 m/s. This explains why overtaking feels slow—from the faster car's perspective, the slower car appears nearly stationary.

Analogy - The Escalator: Standing still on an escalator moving at 1 m/s means you move at 1 m/s relative to ground but 0 m/s relative to the escalator. Walk up at 0.5 m/s, and you're 1.5 m/s relative to ground but still 0.5 m/s relative to the escalator steps.

Example 1: One-Dimensional Relative Motion

Question: Car A travels east at 20 m/s. Car B travels east at 15 m/s. Calculate the velocity of A relative to B.

Solution:

Step 1: Identify the equation: v⃗_AB = v⃗_A - v⃗_B_

Step 2: Define positive direction (east = positive)

• v⃗_A = +20 m/s
• v⃗_B = +15 m/s

Step 3: Calculate v⃗_AB = 20 - 15 = +5 m/s east_

Examiner Note: Always state direction in your final answer (1 mark typically allocated for this).

Example 2: Two-Dimensional Relative Motion

Question: A boat heads north at 8 m/s in water flowing east at 6 m/s. Find the boat's velocity relative to the riverbank.

Solution:

Step 1: Draw a vector diagram with perpendicular components

Step 2: Use Pythagorean theorem for magnitude |v⃗| = √(8² + 6²) = √(64 + 36) = √100 = 10 m/s

Step 3: Calculate direction using trigonometry tan θ = opposite/adjacent = 6/8 = 0.75 θ = tan⁻¹(0.75) = 36.9° east of north

Final Answer: 10 m/s at 36.9° east of north

Examiner Note: Always include both magnitude AND direction (2 marks). Sketch a diagram even if not required—it prevents sign errors.

Example 3: Opposite Directions

Question: Train A travels west at 25 m/s, train B travels east at 20 m/s. Find velocity of A relative to B.

Solution: West = positive v⃗_AB = (+25) - (-20) = +45 m/s west_

Key Point: Opposite directions mean velocities add in magnitude!