What is the longest nap that B can take and still not lose the race?

Question

In a race over a distance s, runner A starts from rest and accelerates at a_1 for the first distance x and then runs at constant speed. Runner B starts from rest and accelerates at a_2 for the first distance x and then runs at constant speed. Runner A begins running as soon as the race begins but B first takes a nap to rest up.

What is the longest nap that B can take and still not lose the race?

Answer 1

The mixed suffixes A,B,1,2 are too confusing; I'll just use a,b.
For runner q in [a,b], the time Tq taken by acceleration Aq over distance x is
Tq = sqrt(2 * x / Aq)
velocity Vq reached is
Vq = Aq * Tq
Time for the whole race TRq is
TRq = Tq + (s - x) / Vq
The allowable nap time is TRa - TRb.