Reference frames and relative motion

Every measurement of position and velocity depends on the observer's reference frame — the coordinate system from which observations are made. The same motion can therefore be assigned different position and velocity values by observers in different frames.

Key Definition An inertial reference frame is one that moves at constant velocity (zero acceleration), and all inertial frames are equally valid for describing motion.

To convert a velocity measurement from one frame to another, use vector addition in one dimension:

• Chain the subscripts so that the inner labels match, allowing the shared middle frame to cancel.
• Assign a sign to each velocity based on a chosen positive direction.
• Add the signed values to obtain the resultant velocity in the new frame.

$v_{\mathrm{AC}}$ = $v_{\mathrm{AB}}$ + $v_{\mathrm{BC}}$

For example, a person walking forward on a moving bus has a larger ground-frame speed than their bus-frame speed, because both velocities point in the same direction and add together: $v_{\mathrm{PG}}$ = $v_{\mathrm{PB}}$ + $v_{\mathrm{BG}}$.

Acceleration in Inertial Reference Frames

Despite the frame-dependence of velocity, acceleration is identical in every inertial reference frame. Adding a constant velocity between frames shifts all velocity values equally, leaving the rate of change of velocity unchanged.

This invariance means Newton's second law applies identically in any inertial frame, and forces measured by different inertial observers always agree.

On the AP exam, relative velocity problems are restricted to one dimension, so careful sign conventions and subscript chaining are the essential tools for avoiding errors.