relative motion reference frames
Motion looks different depending on whose eyes you see it through. A person walking inside a moving train seems to move slowly to fellow passengers and very fast to someone outside. Ride along with one point and look at another, and that point's velocity splits as vB = vA + vB/A: the velocity of the reference point plus the relative velocity seen from it. For two points on the same rigid body, the relative velocity is set by a single rotation, and if the frame itself rotates, a sliding term and a Coriolis term are added on top of ω×r. This lesson builds up, step by step, how to solve for velocity and acceleration by moving from frame to frame.
When two bodies A and B each move on their own, an observer riding on A sees B move with vB/A = vB − vA, the difference of the two velocities. The figure draws the blue arrow vA and the orange arrow vB from one point, and the gold arrow joining their tips is the relative velocity. Change the direction of vB with the slider and the gold arrow changes length and direction with it, its size shown in the readout. Two cars driving side by side in the same direction have a small relative velocity and seem to stand still to each other, while cars driving toward each other have a large one, just as the figure shows.
For two points on the same rigid body, the relative velocity of B is nothing but a pure rotation perpendicular to the rod, ω×rB/A. The rod's length cannot change, so the distance between the two points cannot grow or shrink, and the only relative motion left is turning about A. So vB = vA + ω×r: the diagonal formed by adding the rotational component to the velocity of A is the velocity of B. Increase the angular velocity ω with the slider and the upward rotational component at the end of the rod grows, tilting the orange arrow vB, the sum of it and the copy of vA drawn dashed, upward.
Acceleration carries over the same way, but it splits into two pieces. Pin a rod at A and turn it, and its end receives both a tangential acceleration α×r (perpendicular to the rod) and a centripetal acceleration ω²r (inward along the rod). Increase the angular velocity with the slider and the blue centripetal arrow swells with the square of the angular velocity, while the orange tangential arrow keeps its length, since the angular acceleration is unchanged. Comparing the two numbers in the readout, you can check that doubling the angular velocity makes only the centripetal component four times larger.
When the frame itself rotates, one more term appears. If a bead slides outward along a rod turning at constant angular velocity, the bead's absolute velocity is the rod's rotation ω×r plus the sliding along the rod, vrel. That vrel, the sliding measured on the turning rod, is the new contribution that a rotating frame brings. Move the bead outward along the rod with the slider and the rotational component lengthens in proportion to the radius while the sliding component stays the same, so the orange sum arrow leans more and more toward perpendicular to the rod.
Sliding in a rotating frame adds a term not only to velocity but also to acceleration: the Coriolis acceleration (2ω×vrel). An object sent straight outward from the center of a turning disk, seen from the disk, leaves the dashed straight path and curves sideways like the gold curve. The Coriolis acceleration acts perpendicular to the direction of sliding, so the orange arrow pushing the object sideways is always at right angles to the curve's direction of travel. Increase the angular velocity with the slider and both the amount of curving and the Coriolis acceleration in the readout grow. On Earth, the same effect bends winds and ocean currents to the right in the Northern Hemisphere and to the left in the Southern.