Predicting the future value using the past value

[KK]

Let us assume that a body is to travel from point A to point E, with
B, C, D being intermediate points. The points between A and E are
equally spaced apart by a distance of 10 km. The body is SUPPOSED to
travel AB, BC, CD, DE with speeds of 10 kmph, 20 kmph, 30 kmph, 40
kmph respectively.
With the data given above, let us suppose that the body begins its
journey from A. Please note that the instantaneous speed of the body
is available. Once the body covers AB, we can compute the actual
average speed. Let us suppose that the body actually travels AB with
an average speed of 8 kmph. So, the fact is that the body is SUPPOSED
to cover AB with an average speed of 10 kmph but actually performed
its journey with average speed of 8 kmph. Now the body being
traversed AB, is supposed to traverse BC. At this point can we
PREDICT by how much difference in average speeds (SUPPOSED ~ ACTUAL)
the body can cover BC?
To put it simply, is there a way we can take the history (difference
in average speeds of AB) into account in predicting the future
(difference in average speeds of BC, CD and DE)? Please assume that
the reasons for the difference in actual and Supposed speeds are
completely arbitary.

 

[kuruman]

When you say

... the body is SUPPOSED to cover AB with an average speed of 10 kmph but actually performed its journey with average speed of 8 kmph ...

you are implying that you have some kind of mathematical model that allows you to predict the average speed of the object over equal space intervals. This model must have some parameters the values of which allow you to calculate the average speed (dependent variable) as a function of space interval (independent variable). Obviously, you didn't pick the initial values of the parameters out of thin air, but you made some a priori intelligent guesses and estimates which turned out to be off the mark in retrospect. So you need to take your actual measurements of the instantaneous speed, superimpose the model and adjust the values of the parameters until the model fits the data. Using the values of the parameters from interval AB, you should be able to predict the instantaneous speed for BC. When you do this, there could be three outcomes
1. The new data are consistent with the mathematical model in which case you congratulate yourself for a job well done and proceed to the next step and predict the instantaneous speed over CD.
2. The data are inconsistent with mathematical model to within experimental error in which case you may have to try and refine the values of your parameters by refitting to the data from both space intervals and or by introducing an additional parameter the physical significance of which you can justify.
3. The data and the model are clearly mismatched. This is an indication that you have to abandon your current mathematical model and cook up another one that fits all the data you have so far from AB and allows you to make new predictions for the average speeds that the object is SUPPOSED to have over each interval.

BTW, what I say above is nothing new, just how one proceeds scientifically. You need math to guide your thinking about how nature works and experiments to show how nature actually works and keep the math on track.