A1385 kg cannon containing a 58.5 kg cannon ball is on wheels. the cannon fires the cannon ball giving it a velocity of 49.8 m/s north. what is the initial velocity of the cannon the instant after it fires the cannon ball?

Conservation of momentum requires that the sum of momenta after is equal to that before.  Since initially nothing is moving, the sum after the shot will also add to zero. 
m₁v₁ = -m₂v₂

Solve for the cannon's velocity v₁

v₁ = -m₂v₂/m₁ = -2.10m/s

The negative sign means it's moving 2.10m/s south.

Is a  vector quantity  that refers to "the rate at which an object changes its position." imagine a person moving rapidly - one step forward and one step back - always returning to the original starting position. while this might result in a frenzy of activity, it would result in a zero velocity. because the person always returns to the original position, the motion would never result in a change in position. since velocity is defined as the rate at which the position changes, this motion results in zero velocity. if a person in motion wishes to maximize their velocity, then that person must make every effort to maximize the amount that they are displaced from their original position. every step must go into moving that person further from where he or she started. for certain, the person should never change directions and begin to return to the starting position.