As I understand you have this scenario (just for 3 points)
The green lines represent 20 minute intervals. So in this fictitious example they have walked a shorter distance between 1h-2h as they are going uphill and then a longer distance between 2h-3h as they are going downhill.
So you could use a combination of tools and build a model but I suspect this will be easier to script because you are building, manipulating and extrapolating along individual line segments.
Another (bigger) problem you are going to have is that you are using quite an old version of ArcMap, if you upgrade there will be many new tools and arcpy functions available that could help you do this, for example there is the Generate Points Along Lines tool that is currently not available to you. Another helpful tool which is not available to you is points to lines.
You will make your life considerably easier and open up new processing opportunities if you upgrade...
As a side note it is useful to include simple images/screenshots of what you are trying to achieve, this will attract attention to your question as it often clarifies what a wordy paragraph is trying to explain.