Speed is defined as the change in distance divided by the change in time and written as
s = Δ d/ Δ t.
i.e. s = (d2-d1)/(t2-t1)
If possible we take measurements of distance and time from the object's starting position. This means d1 and t1 will be zero
s = (d2-d1)/(t2-t1) = (d2-0)/(t2-0) = d2/t2
As there is now just one distance and time to consider there is no need for numbered subscripts and so we would just write
s = d/t (where d and t are the object's distance and time from its start position).
To illustrate this further consider a car journey from Manchester to London passing through Birmingham and imagine that you only wanted to know what the average speed was through Birmingham.
If the travel time and distance travelled are measured from the start of the journey in Manchester then to calculate the average speed through
s ave= Δ d/ Δ t
Δ d = (d2-d1)
Δ t = (t2-t1)
save = (d2-d1)/ (t2-t1)
However if the travel time and distance travelled are measured from when the vehicle first reaches Birmingham.
Then the initial distance and time at this point will be zero i.e. d1 and t1 are zero
s ave = (d2-d1)/ (t2-t1) simplifies to:
s ave = (d)/(t)
So taking the start position as the reference point simplifies the calculation.
(also it explains why sometimes the Δ symbols are "dropped" from the equations )