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

Start by setting up some variables that represent duration of the various legs

For runner a, the time it takes to run to point x I'll call n
x=.5*a_1*n^2
n=sqrt(2*x/a_1)

the constant velocity from x to s is
a_1*n
so the duration from x to s i'll call m
s-x=a_1*n*m
m=(s-x)/(a_1*n)
or
m=(s-x)/(a_1*sqrt(2*x/a_1))

for runner B, the nap time I'll call k
the duration from start to x I'll call j
x=.5*a_2*j^2
j=sqrt(2*x/a_2)

the duration from x to s I'll call h
since the constant velocity is
a_2*j
then
s-x=a_2*j*h
like above
h=(s-x)/(a_2*sqrt(2*x/a_2))

for B to exactly tie A the duration for both runner's activities would be equal
n+m=k+j+h
since we want to know the maximum k
k k (s-x)/(a_1*sqrt(2*x/a_1))-
sqrt(2*x/a_2)-
(s-x)/(a_2*sqrt(2*x/a_2))

j