Frame of Reference

Before we can say anything meaningful about how a body moves, we must first answer the question "moving relative to what?" A cup sitting on a train table is at rest with respect to the train, but it is moving at, say, 80 km/h with respect to the ground, and moving even faster with respect to the Sun. Motion is therefore never absolute — it always has to be described relative to a chosen frame of reference.

What a frame of reference is. A frame of reference is a coordinate system (an origin O together with a set of mutually perpendicular axes) rigidly attached to some body, together with a clock to measure time. The position of any object is then specified by its coordinates in this system at each instant of time. For motion confined to a straight line — the subject of this chapter — a single axis (commonly the x-axis) with a chosen origin and a chosen positive direction is enough to fix the position of the moving object at any instant.

Inertial frames of reference. A frame of reference in which Newton's first law of motion holds true — that is, a body free of any net external force continues to remain at rest or move with constant velocity — is called an inertial frame of reference. A frame at rest, or one moving with constant velocity relative to another inertial frame, is inertial. The ground is an excellent inertial frame for most everyday purposes (its rotation and orbital motion produce accelerations far too small to notice in typical laboratory-scale experiments), and a train or car moving at constant velocity in a straight line is inertial as well.

Non-inertial frames of reference. A frame of reference that is itself accelerating (speeding up, slowing down, turning, or rotating) relative to an inertial frame is called a non-inertial frame. In such a frame, Newton's laws do not hold in their simple form unless we additionally introduce fictitious (pseudo) forces that have no real physical origin — they arise purely because the frame itself is accelerating. A commonly experienced example is a bus that brakes suddenly: passengers standing inside feel thrown forward even though no real forward force acts on them; this apparent forward force is a pseudo-force that exists only in the (non-inertial, decelerating) frame of the bus. Similarly, a person standing on a rotating platform feels an outward pseudo-force (the "centrifugal" force), which again has no real agent producing it — it is purely an artefact of describing motion from within the rotating, and hence non-inertial, frame.

Why this matters for straight-line motion. Throughout this chapter, unless stated otherwise, we take the ground (or a laboratory fixed to the ground) as our inertial frame of reference, and we describe the motion of an object as a single coordinate $x$ measured along a straight line from a fixed origin $O$, with a definite choice of positive direction. Every quantity we define next — position, path length, displacement, velocity and acceleration — is measured with respect to this frame. Changing the frame (for example, describing the same motion from inside a moving car instead of from the ground) can change the numerical values of velocity and displacement, which is exactly the idea explored later in Section 2.9 on relative velocity.