Reference Frames and Relative Motion

This topic is about who is doing the measuring. A reference frame is a coordinate system attached to an observer. Velocity depends on the frame: from the platform, Train A moves at $+20\hat{\text{ı}}$ m/s and Train B at $+25\hat{\text{ı}}$ m/s. Step inside Train A and the platform drifts past at $-20\hat{\text{ı}}$ m/s while Train B closes at only $+5\hat{\text{ı}}$ m/s. Same instant, three observers, three sets of numbers — all right. The chain rule that links them is $\vec{v}_{A,B} + \vec{v}_{B,C} = \vec{v}_{A,C}$; inner subscripts cancel. Acceleration is frame-invariant between inertial frames: differentiating the velocity transformation drops the constant relative velocity, so two inertial observers agree on the acceleration of any object even when they disagree on its velocity.

Nearly every error here traces back to treating velocity as if it had no frame — reading Train A's $+20$ m/s as the one correct answer, even though Train A moves at only $-5$ m/s in Train B's frame. From that root grow the rest: holding velocity to be frame-independent, confusing speed with velocity, and botching the signs or subscripts in the chain rule $\vec{v}_{A,B} + \vec{v}_{B,C} = \vec{v}_{A,C}$.