Objects can move along any path in 3 dimensions (e.g a path that combines some forward/backward, left/right and up/down movement). However we will restrict our analysis initialy to objects which are moving in just one dimension. i.e. motion along a straight line. This results in the following simplifications.
Example 1
In 10 seconds an object travels 20 meters from a reference point
Over this period
total distance = 20m
Total displacement is +20m
The magnitude of the displacement is equal to the distance so speed will be equal to the magnitude of velocity
(s = 20/10 = 2 ms^{-1}, v = 20/10 = 2 ms^{-1})
Example 2
In 5 seconds an object travels 2 meters from a reference point before changing direction and then moving 3 metres in the opposite direction
Over this period
total distance = 5m
Total displacement is -1m (2-3)
The magnitude of displacement is not equal to the distance so speed will not be equal to the magnitude of velocity
(s = 5/5 = 1 ms^{-1}, v = -1/5 = -0.2 ms^{-1})
We assume there can be no change of direction during the very short interval used to calculate and instantaneous values
Therefore for one dimension motion the magnitude of instantaneous velocity = instantaneous speed
(note : even though the speed is equal the magnitude of the velocity it still differs from velocity as it has no information about direction)
e.g. if instantaneous velocity is +3 ms^{-1} speed is 3 ms^{-1}.
However if velocity is -3 ms^{-1} speed is still 3 ms^{-1} )