3.3 Relative Motion and Frame of Reference

Frames of Reference and Relative Motion

Motion in Reference Frames

A frame of reference is a coordinate system you attach to some object or location so you can measure position, velocity, and acceleration from that viewpoint. Every time you describe motion, you're describing it relative to a chosen frame.

There are two types of frames you need to know:

• Inertial frame: moves at constant velocity or stays at rest. No net force acts on the frame itself. Earth's surface is treated as approximately inertial for most introductory problems.
• Non-inertial frame: accelerates or rotates. A merry-go-round or a braking car are classic examples. In these frames, you'd feel "fictitious" forces (like being pushed outward on a spinning ride).

Galilean relativity says the laws of mechanics work the same way in every inertial frame. A ball dropped inside a smoothly cruising airplane obeys the same equations as a ball dropped in a lab on the ground. This is why you can't tell you're moving at constant velocity just by doing physics experiments inside a closed room.

Relative Velocity

The core equation for relative motion is:

$\vec{v}_{AB} = \vec{v}_{A} - \vec{v}_{B}$

This gives you the velocity of object A as observed by object B. Pay close attention to the subscript order: $\vec{v}_{AB}$ means "A's velocity relative to B."

In one dimension, this simplifies to plain subtraction along a single axis. If car A moves east at 80 km/h and car B moves east at 50 km/h, then A's velocity relative to B is $80 - 50 = 30$ km/h east. If they travel in opposite directions (B heading west at 50 km/h), you'd treat B's velocity as negative, giving $80 - (-50) = 130$ km/h. That's why head-on collisions are so dangerous.

In two dimensions, you need to work with vector components. Here's the process:

1. Set up a coordinate system (choose your positive x and y directions).

2. Break each velocity vector into x and y components using $v_x = v\cos\theta$ and $v_y = v\sin\theta$.

3. Subtract the corresponding components: $(v_{AB})_x = (v_A)_x - (v_B)_x$ and $(v_{AB})_y = (v_A)_y - (v_B)_y$.

4. Recombine using the Pythagorean theorem: $|\vec{v}_{AB}| = \sqrt{(v_{AB})_x^2 + (v_{AB})_y^2}$.

5. Find the direction with $\theta = \tan^{-1}\left(\frac{(v_{AB})_y}{(v_{AB})_x}\right)$.

You can also solve these graphically by drawing the vectors tip-to-tail, but the component method is more reliable for getting exact answers.

Applications of Relative Motion

River-crossing problems are the most common application you'll see. A boat aims straight across a river, but the current pushes it downstream. The boat's velocity relative to the ground is the vector sum of its velocity relative to the water and the water's velocity relative to the ground:

$\vec{v}_{BG} = \vec{v}_{BW} + \vec{v}_{WG}$

If a boat crosses a 200 m wide river at 4 m/s relative to the water, and the current flows at 3 m/s, the boat's ground speed is $\sqrt{4^2 + 3^2} = 5$ m/s, and it lands downstream of its starting point. To land directly across, the boat must angle upstream to compensate.

Airplane-wind problems work the same way. A pilot's airspeed (speed relative to the air) combines with the wind velocity to produce the ground speed. A headwind reduces ground speed; a tailwind increases it; a crosswind pushes the plane off course.

Other situations where relative motion matters:

• A person walking on a moving walkway at an airport has a ground speed equal to their walking speed plus the walkway speed.
• Two trains passing each other: their relative speed is the sum of their individual speeds if they travel in opposite directions.

Choosing the Right Reference Frame

Picking a smart reference frame can turn a complicated problem into a simple one. If two objects are both moving, switching to a frame where one of them is stationary often cuts the number of variables you need to track.

For example, to analyze a package sliding on a moving conveyor belt, choosing a frame attached to the belt lets you treat the belt as stationary and focus only on the package's motion relative to it.

The deeper point here is that there's no such thing as "absolute motion." All motion is measured relative to something. The Sun doesn't truly "move across the sky"; Earth rotates. Both descriptions are valid in their respective frames, but choosing Earth's rotation as your frame gives you a more useful physical picture.