One dimensional relative velocity(in a line)
Consider two particles A and B at instant t positioned along the xaxis from point O.
Particle A has a displacement x_{A} from O, and a velocity V_{A} along the xaxis. The displacement x_{A} is a function of time t .
Particle B has a displacement x_{B} from O, and a velocity V_{B} along the xaxis. The displacement x_{B} is a function of time t .
The velocity V_{B} relative to velocity V_{A} is written,
_{B}V_{A} = V_{B}  V_{A}
This can be expressed in terms of the derivative of the displacement with respect to time.
Two dimensional relative position & velocity
Particle A has a displacement r_{A} from O, and a velocity V_{A} along the xaxis. The displacement r_{A} is a function of time t .
Particle A has a displacement r_{B} from O, and a velocity V_{B} along the xaxis. The displacement r_{B} is a function of time t .
Relative position
The position of B relative to A at time t is given by the position vector from O, r_{BA} .
The position vector r_{BA} can be written as,
_{B}r_{A} = r_{B} r_{A}
Relative velocity
Similarly, at time t the velocity vector V_{B} relative to velocity vector V_{A} can be written,
_{B}V_{A} = V_{B}  V_{A}
This can be expressed in terms of the derivative of the displacement with respect to time.
Example #1
If the velocity of a particle P is (9i  2j) ms^{1} and the velocity of another particle Q is (3i  8j) ms^{1} , what is the velocity of particle P relative to Q?
Example #2
A particle P has a velocity (4i + 3j) ms^{1}. If a second particle Q has a relative velocity to P of (2i  3j), what is the velocity of Q?
Example #3
A radar station at O tracks two ships P & Q at 0900hours (t=0) .
P has position vector (4i + 3j) km, with velocity vector (3i  j) km hr ^{1}.
Q has position vector (8i + j) km, with velocity vector (2i + 2j) km hr ^{1}.
i) What is the displacement of P relative to Q at 0900 hours? (ie distance between ships). Answer to 2 d.p.
ii) Write an expression for the displacement of P relative to Q in terms of time t .
iii) Hence calculate the displacement of P relative to Q at 1500 hours.
iv) At what time are the two ships closest approach and what is the distance between them at this time?
i)
ii)
therefore the displacement of P relative to Q is given by,
iii) using the result above for 1500 hours( t = 6 )
iv) Closest approach is when the position vector of P is at right angles to the reference vector.
The 'reference vector' is the first part of the vector equation for r .
The position vector gives the point P at time t along the straight line described by the vector equation.
(solution to follow)
Two dimensional relative acceleration
Similarly, if a_{A} and a_{B} are the acceleration vectors at A and B at time t, then the acceleration of B relative to A is given by,

