%Project2_part2
%total arc length
tal = 2.495246747676093; % Total arc length
 TOL = 1e-3;
 s = input('input the fraction of the curve you would like to know the time value for: ');
% Bisection Method
% Predefine the begining values for a and b
a=0; b=1; tol=1e-6;
fa = adapquad(f_t,0,a,tol) - s*tal;
fb = adapquad(f_t,0,b,tol) - s*tal;
%function t_star = bisect(f,a,b,tol)
if sign(fa)*sign(fb) >= 0
  error('f(a)f(b)<0 not satisfied!') %ceases execution
end
fa=adapquad(f_t,0,a,tol) - s*tal;
fb=adapquad(f_t,0,b,tol) - s*tal;
while (b-a)/2>tol
  c=(a+b)/2;
  fc=adapquad(f_t,0,c,tol) - s*tal;
  if fc == 0              %c is a solution, done
    break
  end
  if sign(fc)*sign(fa)<0  %a and c make the new interval
    b=c;fb=fc;
  else                    %c and b make the new interval
    a=c;fa=fc;
  end
end
time=(a+b)/2;               %new midpoint is best estimate
Arc_length = adapquad(f_t,0,time,tol)
Arc_length_over_total = adapquad(f_t,0,time,tol)/tal
time