Let’s say you have to run out and back a little way as quickly as you can. It’s only a short distance, so we’ll assume that your ability to accelerate is limited by the power you can put out, and that your ability to decelerate is limited by the friction between your shoes and the surface you’re running on.
You have some set maximum power output (which does not deteriorate). You execute your race perfectly, outputting maximum power until the exact moment you need maximum breaks to come to a halt at the turnaround point. You decelerate as hard as you can, and when you’ve stopped at the turnaround, you accelerate at maximum power again all the way through the line (you don’t need to stop at the finish line this time – you can run through).
How does the amount of time it takes you to run the race scale with your maximum power output?