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- Motion in a Plane - Relative Velocity

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## Relative Velocity

The relative velocity of an object A with respect to object B is the rate of change of position of the object A with respect to object B.

If V_{A} and V_{B} be the velocities of objects A and B with respect to the ground, then

The relative velocity of A with respect to B is V_{AB} = V_{A} – V_{B}

The relative velocity of B with respect to A is V_{BA} = V_{B} – V_{A}

**Relative Motion in One Dimension**In one-dimensional motion, objects move in a straight line. So there are only two possible cases:

Objects are moving in the same direction

Objects are moving in the opposite direction

Again take the example of a man sitting on the train if the train is moving with 100 km/hr forward. Then according to the man sitting on the train, the trees outside are moving backwards with 100 km/hr.

Because from the man’s point of view, the outside environment is moving in the opposite direction to the train with the same velocity.

So for all types of questions, if you have to find the velocity of A with respect to B, then assume that B is at rest and give the velocity of B to A in the opposite direction.

**Relative Motion in Two Dimensions**The same concept will be applicable in two-dimensional motion. If you have to find the velocity of A with respect to B, assume that B is at rest and give the velocity of B to A in the opposite direction.

Let us consider two objects A and B which are moving with velocities V_{A} and V_{B} with respect to some common frame of reference, say, with respect to the ground or the earth. We have to find the velocity of A with respect to B, so assume that B is at rest and give the velocity of B to A in the opposite direction.

V_{AB} = V_{A} –V_{B}Similarly, for the velocity of object B with respect to A, assume that A is at rest and give the velocity of A to B in the opposite direction.

V_{BA} = V_{B} – V_{A}

V_{AB} = - V_{BA}

|V_{AB}| = |V_{BA}|