Motion relative to moving frames

Two observers may provide different description of the same event if they are positioned in different frames. For example, a stationary point on the ground may look to an observer stationed on a merry-go-round as moving. Or, for example, passengers sitting in an airplane may consider each other as stationary, while an observer on earth may consider them to be moving.

Frames: To describe position in space we use a frame. The frame we use is frequently the background of the event we are observing, yet we can select any rigid body as our frame to reference our observations. The earth is the most common frame we use, but it is not the only one. For example, observations in the shuttle may be done relative to the “shuttle frame,” and then translated to earth frame.

Time derivatives and frames: The derivative of any variable with respect to time provides information on how it is changing with time. Since the same event when viewed from different frames may result in different observations, the time derivative of an event when taken relative to one frame may be different from the time derivative taken relative to another frame. For example, point A which is fixed to frame  has zero velocity as seen by an observer on frame , yet will have a downward velocity of  if observed by an observer sitting at point O on the frame , which is a disk rotating about O with angular velocity  relative to .

One can relate the time derivative obtained from an observer in one frame to that obtained from an observer in another frame. Let the vector  be represent a quantity being observed by two observers, one on frame  and one on frame . Let denote the derivative of obtained by the observer on , and let  denote the derivative of obtained by the observer on .  These derivatives are related by following equation

where is the angular velocity of frame  with respect to frame .

Relating observations on velocity and acceleration: The position of a point P as it moves on its path can be described relative to the origin O of frame  by the position vector  or relative to the origin A of frame  by the position vector . These vectors are related through the law of vector addition given by

Taking derivatives of both sides of the equation from the point of view of the observer in  and application of the above relation to relate derivatives in two different frames results in the relation

The first term on the right hand side is the velocity as seen by an observer sitting at point A on frame , the second term on the right hand side is a modification of this velocity to account for the rotation of frame  with respect to , and the third term on the right hand side is a modification of the first term to account for the translation of  with respect to . The derivative of this gives the relation between the acceleration as observed from the two frames. After application of the rule for taking derivatives and some reorganization one gets

The first term on the right hand side is the acceleration measured by an observer located at point A  on frame , the next three terms are modifications to correct for the rotation of frame   with respect to , and the last term is to correct for the translation of  with respect to .

Note: The derivative of is the same in both frames. This follows from

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Department of Engineering Mechanics, University of Nebraska, Lincoln, NE 68588-0526