Reference frames and relative motion

What Is a Reference Frame?

Every measurement in physics depends on who is watching and where they are standing. A reference frame is a coordinate system attached to a specific observer, used to measure position, velocity, and other quantities. The choice of reference frame determines both the direction and magnitude of every measured quantity.

Think of riding in a car on a highway. A passenger in your car sees you as stationary. A person standing on the roadside sees you moving at 30 m/s. Neither observer is wrong — they simply measure from different reference frames. The physics is the same, but the numbers change.

MisconceptionStudents often believe one reference frame is “correct” and others are wrong. All inertial frames are equally valid for describing motion.

An inertial reference frame is one that moves at constant velocity (including zero). In such a frame, Newton’s laws hold without modification. A frame that accelerates is non-inertial. Unless a problem states otherwise, assume the reference frame is inertial.

BoundaryUnless otherwise stated, the frame of reference of any problem may be assumed to be inertial. Relative velocity problems on the AP Physics 1 exam are restricted to one dimension only.

This is outside the scope of the AP exam.

Exam TipIf asked to describe motion, always state which observer or frame you are using.

Converting Between Reference Frames

Key Equations

Relative velocity (one dimension):

$${v}_{AC}={v}_{AB}+{v}_{BC}$$

Variables:

• ${v}_{AC}$ = velocity of object A relative to frame C
• ${v}_{AB}$ = velocity of object A relative to frame B
• ${v}_{BC}$ = velocity of object B relative to frame C

SI unit: m/s

Rearrangements:

$${v}_{AB}={v}_{AC}-{v}_{BC}$$

$${v}_{BC}={v}_{AC}-{v}_{AB}$$

ProportionalityThe observed velocity is a linear sum — no squared or inverse relationships apply here.

Reference sheet status: This equation is not on the AP reference sheet. Know the subscript-chaining method.

The subscript notation follows a chain rule: the inner subscripts must match. In ${v}_{AC}={v}_{AB}+{v}_{BC}$, notice that B appears as the second subscript of the first term and the first subscript of the second term. This chaining always works and prevents sign errors.

Measurements made in one reference frame can be converted to another frame using vector addition. In one dimension, this means adding or subtracting velocities with careful attention to sign.

Choosing a sign convention: Before any calculation, define a positive direction. For example, “taking east as positive.” Any velocity in the opposite direction gets a negative sign. This is essential because combining velocities in one dimension is really adding signed numbers, not just magnitudes.

Consider a person walking forward inside a moving train. Relative to the train, the person moves at ${v}_{PT}$. Relative to the ground, the train moves at ${v}_{TG}$. The person’s velocity relative to the ground is:

$${v}_{PG}={v}_{PT}+{v}_{TG}$$

If the person walks in the same direction the train moves, both terms share the same sign and the ground-frame speed is larger. If the person walks opposite to the train’s motion, the terms have opposite signs and partially cancel.

Examiner InsightAP FRQs often present relative motion scenarios and ask students to explain why two observers disagree on a velocity. Always identify both frames and state the velocity of one frame relative to the other.

Exam TipUse subscript chaining to set up the equation before substituting numbers.

Acceleration Is the Same in All Inertial Frames

Velocity measurements change between inertial reference frames, but acceleration does not. Every inertial observer measures the same acceleration for a given object. This is a powerful and often surprising result.

The reason is straightforward. When converting between two inertial frames, you add a constant velocity. A constant added to every velocity value does not change how quickly velocity changes. Because acceleration is the rate of change of velocity, adding a constant velocity shifts every velocity reading by the same amount. The differences between successive velocity values stay the same, so the acceleration is unchanged.

This means Newton’s second law gives identical results in every inertial frame. The net force on an object, its mass, and its acceleration are all frame-independent quantities. Only position and velocity depend on the observer’s frame.