Reference frames and relative motion

Every kinematic measurement — position, velocity, displacement — depends on the observer's reference frame.

Key Definition A reference frame is the coordinate system attached to an observer, and no single inertial frame is more correct than any other.

When two inertial frames move relative to each other, the velocity of an object transforms by vector addition: the observed velocity equals the object's velocity in one frame plus the relative velocity between the frames.

The subscript chain method organizes relative-velocity calculations cleanly for both one-dimensional and two-dimensional (vector) problems.

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

• The inner (adjacent) subscripts must match — here B — while the outer subscripts give the result, the velocity of A relative to C.
• Reversing the order of the subscripts reverses the vector: $v_{\mathrm{BA}}$ = -$v_{\mathrm{AB}}$.

Acceleration Invariance

While velocities and displacements differ between inertial frames, acceleration does not.

Because inertial frames differ only by a constant relative velocity, the time derivative of that constant vanishes, leaving the acceleration unchanged.

• Consequently, Newton's second law, forces, and masses are identical for all inertial observers.

On the AP exam, stating and using this invariance is essential whenever a problem involves multiple observers or moving platforms.