4.11:
Relative Velocity in Two Dimensions
Consider a woman watching an airplane taking off. As the plane moves farther away, the relative motion in the reference frame of the woman can be determined by velocity vector diagrams.
In another example, a boat and a ship are traveling apart from each other with an angle of 45°. When observed from the moving ship, the boat appears to move away at an angle higher than 45°.
The velocity of the boat relative to water is 35 kilometers per hour and the ship's velocity is 40 kilometers per hour. What will be the velocity of the boat relative to the ship?
Initially, list the known and unknown quantities. Label the vector triangle sketch. Make sure the sum of the angles of the triangle is equal to 180°.
Using the law of sines and by substituting the magnitude of the sides and their opposite angles of the triangle, the velocity of the boat relative to the ship is calculated.
4.11:
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.