Reference Frames

Every measurement of position, velocity, or acceleration is made relative to something. That something is a reference frame: a coordinate system attached to an observer, usually the ground, but it could just as easily be a moving train, car, or airplane. When you switch frames, you are not changing the object's actual motion, you're changing the ruler and stopwatch you use to describe it. Velocities add simply when frames move relative to each other in a straight line: the object's velocity in the ground frame equals its velocity relative to the moving frame plus that frame's velocity relative to the ground.

This is why a ball dropped inside a moving train looks different to different people. To a passenger, the train frame, the ball has no sideways velocity, so it just falls straight down. To a bystander on the platform, the ground frame, the ball already shares the train's forward velocity before it's dropped, so it follows a curved path, forward and down together. Both descriptions are correct. There is no single frame that sees the 'real' velocity, because velocity is never an absolute property of an object, only a relationship between the object and whoever is watching.

This idea sets up everything else in kinematics: when a problem gives you a velocity, it is always implicitly relative to some frame, usually the ground unless stated otherwise. It also previews why projectile motion works the way it does, and later, why the laws of physics themselves, not just the numbers, look the same in every non-accelerating frame.