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Velocity and acceleration of a particle can be expressed in different reference frames. If frame B moves relative to frame A, then v_A = v_B + v_B/A and a_A = a_B + a_B/A + 2ω × v_B/A (if B rotates). This allows analysis of complex motions by choosing convenient reference frames.
From your earlier kinematics work, you know how to describe the motion of a single particle — its position, velocity, and acceleration in rectilinear and curvilinear coordinates. All of that assumed a fixed observer. Relative motion analysis asks: what happens when the observer is also moving? This is not just an abstract question. In engineering practice, a passenger on a moving train, a valve inside a spinning turbine, or a component sliding along a rotating robotic arm all require you to relate observations made in a moving frame to the fixed-frame coordinates you ultimately care about.
The core idea for non-rotating frames is additive: v_P/A = v_P/B + v_B/A. The velocity of particle P with respect to fixed frame A equals the velocity of P measured inside moving frame B, plus the velocity of frame B itself. The translation law applies to position and acceleration equally. Think of it as vector addition: if you're walking at 2 m/s relative to a train that's moving at 30 m/s relative to the ground, your ground-frame velocity is 32 m/s (or 28 m/s if you walk backwards). Each measured velocity is a vector, so direction matters.
Rotating frames add a subtlety: even if a particle is stationary inside the rotating frame, it has acceleration in the fixed frame simply because the frame itself is spinning. The full acceleration transformation is a_A = a_B + α × r_P/B + ω × (ω × r_P/B) + 2ω × v_P/B (rel). The last term, 2ω × v_P/B (rel), is the Coriolis acceleration — it arises because a particle moving inside a rotating frame has its direction changed by the rotation, creating an additional acceleration even at constant speed relative to the frame. Intuitively: if you walk radially outward on a spinning carousel, you are constantly entering faster-moving lanes of the carousel, and the carousel must push you sideways to accelerate you up to each new lane's speed. That sideways push is the Coriolis effect.
Choosing the right reference frame is an engineering strategy. If a problem involves a particle constrained to slide along a rotating rod, the motion relative to the rod is simple (pure translation along one axis), while the absolute motion is complicated. Attaching frame B to the rotating rod makes the relative-motion part trivial and reduces the problem to bookkeeping the frame's own rotation. This decomposition — break complex absolute motion into simple relative motion plus simple frame motion — is the practical value of the reference-frame approach. For mechanisms with multiple links, you chain frames together, applying the transformation at each joint. The builds-toward topics (rigid body plane motion, constrained particle motion) extend exactly this strategy to extended bodies rather than particles.
A common trap is mixing derivatives taken in different frames. The velocity of a point measured in frame B (its time rate of change of position as seen by an observer fixed in B) is not the same as the time derivative of position taken in frame A, even if the two observers agree on the current position vector. Always be explicit about which frame each derivative is taken in before adding or subtracting velocity vectors.