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Typically, relative motion problems involve either two points, say A and B, moving independently of each other, or they involve two points on the same rigid body. The equations needed look the same. Also, relative motion equations can be written for “position”, “velocity”, or “acceleration” of the two points, though velocity and acceleration problems are the most common.
The most confusing aspect of these equations is their form and notation. Fortunately, the form and notation provided in the Reference Handout is what I consider the least confusing form and notation. The absolute position, velocity, or acceleration of one of the points, labeled with a subscript A, is set equal to the absolute position, velocity, or acceleration to the other point, labeled with a subscript B, plus a term with the subscript A/B. This second term is the relative motion term. Note that the labeling of points A and B can be reversed, meaning the position, velocity, or acceleration of point B is set equal to the position, velocity, or acceleration of point A, plus a term with the subscript B/A. So you can either have, in terms of subscripts, A = B + A/B, or B = A + B/A. It is this second term that is the most confusing.
Taking advantage of this notation, and limiting our discussion to velocity or acceleration, the term with the A/B subscript means the velocity or acceleration of point A “relative” to point B “as it B is fixed.” Point B is not fixed, but for this term it is treated as fixed. This means the term with the subscript B/A means the velocity or acceleration of point B relative to point A “as if A is fixed.” And if the two points are on the same rigid body, the motion of one point about another is circular motion. This means the second or “relative” term will contain the angular velocity or angular acceleration of the rigid body. It will either be given or it will be unknown.
The last key issue of the relative motion equations is that they are “vector” equations. So for 2D Plane motion, the single vector equation will become two scalar equations. These two equations will be solved simultaneously as they typically contain two unknowns.
Therefore, solving relative motion problems starts by appropriately choosing and labeling two points, separate the vector equation into scalar equation, evaluate the relative motion term, and lastly solve the scalar equations for the unknowns. Doing these steps in this order will minimize any confusion on how the body moves. – Dr. Tom
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