Relative Velocity in Two Dimensions

Relative velocity is the velocity of an object as observed from a particular reference frame, or the velocity of one reference frame with respect to another reference frame. The concept of relative velocity can be used to describe motion in two dimensions. Consider a particle P and two reference frames S and S′. The position of the origin of S′ as measured in S is , the position of P as measured in S′ is , and the position of P as measured in S is , which can be evaluated by utilizing vector addition as 

Also, the relative velocities are the time derivatives of the position vectors. Therefore,

The velocity of a particle relative to S is equal to its velocity relative to S′ plus the velocity of S′ relative to S. This can be extended to any number of reference frames.

For particle P with velocities , and in frames A, B, and C, the relation can be stated as

Also, the relationship between the accelerations observed in two reference frames can be obtained by differentiating the velocity equation

This text is adapted from Openstax, University Physics Volume 1, Section 4.5: Relative Motion in One and Two Dimensions.