Relative Motion

Orientation

Lesson goal: build accurate physics fluency for relative motion and use that fluency to support clear HSC-style scientific writing.

This page is materialised into the MentorMind course shell from existing teaching, textbook, and eduKG material. Use it as the main lesson surface; use the tutor for targeted repair, worked examples, and concise writing feedback.

Syllabus inquiry question

• How is the motion of an object moving in a straight line described and predicted?

From The Feynman Lectures on Physics, Vol I, Chapter 15:

Motion is always measured relative to a chosen frame. Changing the frame changes the numbers, but it does not change the physics.

Learning Objectives

• Define relative position and relative velocity.
• Solve one-dimensional relative motion problems.
• Apply vector subtraction to two-dimensional relative motion.
• Interpret motion from different reference frames.

Content

Relative velocity in one dimension

Relative velocity compares two objects moving along the same line:

$$v_{AB} = v_A - v_B$$

This reads as "the velocity of A relative to B".

$v_{AB}$ means "velocity of A as observed from B" or "velocity of A relative to B"

Interactive: One-Dimensional Relative Motion

Consider two cars on a highway. Their relative velocity determines how quickly the gap between them changes.

Example: Car A at 25 m/s, Car B at 18 m/s (both east)

• Relative velocity: $v_{AB} = 25 - 18 = 7$ m/s
• Car A approaches Car B at 7 m/s

Relative velocity in two dimensions

Vectors are subtracted component-wise. The relative velocity points from the observer to the object being described.

$$\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$$

Interactive: Boat and River Current

A classic relative motion problem: a boat crossing a river with current.

Interpreting the diagram:

• Blue vector: boat's velocity relative to water (heading north)
• Green vector: water's velocity relative to ground (current flowing east)
• Red dashed vector: boat's velocity relative to ground (resultant)

Reference frames

A statement about motion is incomplete without a frame. The same motion can be described differently in different frames without contradiction.

All velocity measurements require specifying:

1. What is moving
2. Relative to what it's being measured

Common Relative Motion Scenarios

Worked Examples

Example 1: Overtaking cars

Car A travels at 25 m/s east. Car B travels at 18 m/s east. Find A relative to B.

Solution:

1. $v_{AB} = v_A - v_B = 25 - 18 = 7$ m/s
2. The positive result indicates A moves east relative to B
3. A closes the gap at 7 m/s

Example 2: Boat and current

A boat heads due north at 4.0 m/s relative to the water. The current is 1.5 m/s east. Find velocity relative to the ground.

Solution:

1. Components: $v_x = 1.5$ m/s, $v_y = 4.0$ m/s
2. Speed: $\sqrt{1.5^2 + 4.0^2} = \sqrt{18.25} = 4.3$ m/s
3. Direction: $\tan^{-1}(1.5/4.0) = 20 degrees$ east of north

Example 3: Rain and a moving cyclist

Rain falls vertically at 6.0 m/s. A cyclist rides east at 5.0 m/s. Find the rain velocity relative to the cyclist.

Solution:

The rain's velocity relative to the cyclist is found by vector subtraction:

$$\vec{v}{rain/cyclist} = \vec{v}{rain} - \vec{v}_{cyclist}$$

1. Rain (relative to ground): $v_x = 0$, $v_y = -6.0$ m/s (downward)
2. Cyclist (relative to ground): $v_x = 5.0$ m/s, $v_y = 0$
3. Rain relative to cyclist: $v_x = 0 - 5.0 = -5.0$ m/s, $v_y = -6.0$ m/s

Result:

• Speed: $\sqrt{5.0^2 + 6.0^2} = 7.8$ m/s
• Direction: $\tan^{-1}(5.0/6.0) = 40 degrees$ from vertical, toward the cyclist (west of vertical)

The rain appears to come from ahead! This is why cyclists lean forward in rain.

Example 4: Head-on collision approach

Two trains approach each other. Train A travels at 30 m/s east, Train B at 25 m/s west.

Solution:

Taking east as positive:

• $v_A = +30$ m/s
• $v_B = -25$ m/s

Relative velocity:

• $v_{AB} = v_A - v_B = 30 - (-25) = 55$ m/s

The trains approach each other at the sum of their speeds.

Interactive: Aircraft Navigation

An aircraft must aim off-course to compensate for wind:

Navigation problem: To fly due north (90 degrees), the pilot must aim slightly west to compensate for the eastward wind.

Common Misconceptions

• Misconception: Relative velocity adds magnitudes. Correction: It is a vector difference; direction matters.

• Misconception: The faster object always has a positive relative velocity. Correction: The sign depends on the chosen frame and direction convention.

• Misconception: Reference frames change the laws of motion. Correction: They change measurements, not the laws.

• Misconception: Objects moving in the same direction have zero relative velocity. Correction: Only if they have the same speed in the same direction.

Practice Questions

Easy (2 marks)

A bus moves at 12 m/s east. A passenger walks at 1.5 m/s east relative to the bus. Find the passenger's speed relative to the ground.

• Correct relative velocity model (1)
• Correct final speed with units (1)

Answer: $v_{pg} = v_{pb} + v_{bg} = 1.5 + 12 = 13.5$ m/s east

Medium (4 marks)

A swimmer aims straight across a river at 1.8 m/s relative to the water. The current flows 1.2 m/s downstream. Find the swimmer's ground velocity and direction.

• Correct vector addition (2)
• Correct speed and direction (2)

Answer:

• $v_x = 1.2$ m/s (downstream), $v_y = 1.8$ m/s (across)
• Speed: $\sqrt{1.2^2 + 1.8^2} = 2.2$ m/s
• Direction: $\tan^{-1}(1.2/1.8) = 34 degrees$ downstream from straight across

Hard (5 marks)

Plane A flies 250 km/h north. Plane B flies 180 km/h east. Determine the velocity of A relative to B.

• Correct vector subtraction setup (2)
• Correct relative speed (2)
• Correct direction description (1)

Solution:

$\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$

Components:

• $v_{AB,x} = 0 - 180 = -180$ km/h (west)
• $v_{AB,y} = 250 - 0 = 250$ km/h (north)

Relative speed:

• $v_{AB} = \sqrt{180^2 + 250^2} = \sqrt{94900} = 308$ km/h

Direction:

• $\theta = \tan^{-1}(180/250) = 36 degrees$ west of north

Answer: 308 km/h at 36 degrees west of north

From B's perspective, A appears to move northwest.

Multiple Choice Questions

Test your understanding with these interactive questions:

Summary

• Relative motion compares objects using a chosen frame.
• One-dimensional problems use signed subtraction.
• Two-dimensional problems require vector subtraction.
• Clear frame statements prevent sign and direction errors.

Self-Assessment

Check your understanding:

After studying this section, you should be able to:

• Explain what "velocity of A relative to B" means
• Calculate relative velocity in one dimension
• Apply vector subtraction for two-dimensional relative motion
• Solve boat-and-river problems
• Describe motion from different reference frames

Module 1 Complete

Congratulations on completing Module 1: Kinematics!

• Motion in a straight line (displacement, velocity, acceleration)
• Graphical analysis of motion
• SUVAT equations for constant acceleration
• Vector operations in two dimensions
• Relative motion between objects

Scientific Writing And Exam Support

When answering questions from this lesson, separate:

• the physical quantity being discussed,
• the model or law being applied,
• the mathematical relationship, including units,
• the conclusion in words.

For explanation questions, write in the pattern: claim -> physics reason -> consequence. For calculation questions, state the formula, substitute with units, calculate, then interpret the answer.

Maintenance Loop

One-minute retrieval:

1. State the key law, model, or relationship used in this lesson.
2. Identify one common misconception that would lead to a wrong answer.
3. Write one sentence that links the calculation or evidence back to the physical meaning.