Converting measurements between reference frames lets you describe the same motion from different observers’ viewpoints. The key is being consistent about coordinate origins, positive direction, and the relative motion of the frames.

Core idea: converting the “same event” between frames

A measurement (position, displacement, velocity) depends on the reference frame you choose. Two observers can assign different numerical values while still describing the same physical motion, as long as they use a consistent conversion.

What must be specified before converting

• The two frames (often labelled A and B)

• Each frame’s positive direction in 1D

• The relative motion of the frames (constant speed and direction)

• Any offset between coordinate origins at a chosen $t=0$

A frame speed is not the object’s speed; it is the “background” motion you subtract or add to translate one observer’s measurements into the other’s.

Position and displacement conversions (1D, constant frame speed)

In AP Physics 1, conversions are typically Galilean (non-relativistic): time is treated the same for both observers, and coordinates differ by a shift that grows linearly with time if the frames slide past each other.

This equation encodes two ideas: (1) a fixed shift between origins ($x_{B0/A}$), and (2) a changing shift due to relative sliding ($v_{B/A}t$).

The top-left panel presents a Galilean reference-frame sketch: two coordinate systems aligned with a relative translation that grows linearly in time due to constant relative speed. Visually, the same event is located by different coordinate readouts because the origins do not coincide except at the chosen alignment time. This supports interpreting position conversion as “subtract the frame’s drift plus any initial origin offset.” Source

Displacement is often simpler than position

If you compare two events at times $t_1$ and $t_2$, the origin offset cancels. With constant frame speed:

• Convert positions if you need where something is relative to a particular origin.

• Convert displacements if you only need the change in position between two times.

Velocity conversions

Velocity measurements differ between frames by the frame speed (with sign).

Conceptually: the observer in B “removes” the background motion of B relative to A.

Practical sign-check for 1D

• If frame B moves in the + direction relative to A, then $v_{B/A}$ is positive.

• If an object’s velocity in A equals the frame speed, then the object is at rest in B (its converted velocity becomes zero).

• If your converted velocity flips sign unexpectedly, re-check the chosen + direction for each frame and keep it consistent.

A reliable conversion workflow (no calculation tricks)

• Choose one axis direction (e.g., +x to the right) and stick to it in both frames.

• Write down $v_{B/A}$ with a sign.

• Decide whether you need:

  • a position conversion (needs origin offset information), or

  • a displacement/velocity conversion (often does not).

• Check limiting cases:

  • If $v_{B/A}=0$, both frames must give the same measurements.

  • If the object is stationary in A, then in B it should move opposite the direction that B moves relative to A.