Reference Frames

Every position, velocity, and acceleration you write down is measured relative to some frame, a coordinate system that may itself be moving. Drop a coin inside a train moving at constant velocity and it falls straight down as far as you're concerned, because you and the coin share the train's sideways velocity and it cancels out in your view. Someone standing on the platform sees the coin start with the train's forward velocity and then fall, so its path curves into a slight arc. Both descriptions are correct. They're just using different frames.

To move a velocity from one frame to another, add velocities as vectors. If an object moves with velocity v_object/B relative to frame B, and frame B itself moves with velocity v_B/A relative to frame A, then the object's velocity relative to A is v_object/A = v_object/B + v_B/A. This is just vector addition, so you break each velocity into components, add component by component, and recombine if you need a magnitude and direction. The same rule runs in reverse: subtract a velocity to switch which frame you're viewing from.

This matters because Unit 1 treats velocity and acceleration as derivatives of position, and derivatives depend on which position function you're differentiating — the one measured in frame A or the one in frame B. Switching frames shifts velocity by a constant (the relative velocity between frames), but if that relative velocity is constant, it does not shift acceleration at all: both observers agree on acceleration, and therefore on the forces causing the motion. That agreement is what makes an inertial frame special, and it's the idea this topic hands off to Newton's laws, where force and acceleration must match for every inertial observer.