## Relative Velocities and Relative Positions

Definition: If A is at positionmoving with velocityand B is at positionmoving with velocitythen

The relative position of B relative to A is given byand the relative velocity of B relative to A is given by

The relative position of A relative to B is given byand the relative velocity of A relative to B is given by

The position ofof A at any time t is given by

The position ofof B at any time t is given by

Typically we are are asked about relative velocities, relative positions and points of intersection or CRASHES!

Example A boat A started from positionwith speed.

a)Find the position of A at time t

If boat B start from positionwith velocityfind if boat A and B CRASH and if they do, find the position of the crash site and the distance from the origin to this position.

The position vector of B at time t is given by

If they collide, at some point they are in the same place at the same time. This means that for some value of We solve:

We equate the coefficients ofandon both sides:

We get the same value of t from both equations hence they meet at the same place at the same time.

We can obtain the position vector of the point of collision be substituting the value t=2 into eitheror

The distance from the origin tois given byso distance from origin to crash site is