Hmm, I did it differently. First, I used x(t) and y(t) instead of f(t) and g(t), just because it is more descriptive.
We know (if I understand EAT correctly):
y'(t)
2 + x'(t)
2 = c
2 (speed of light is constant)
y''(t) = a (light accelerates upwards constantly with acceleration a)
y(0) = 0, x(0) = 0, y'(0) = 0 (initial conditions)
y''(t) = a => y'(t) = at =>
y(t) = at2/2 This is assuming the EA is constant and directed upwards.
So, by pythagoras: (at)
2 + x'(t)
2 = c
2Re-arrange for x'(t) to get a difficult integral, type into the
online integrator to get:
x(t) = (c2/2a) arctan (at/{c2-(at)2}0.5) + (t/2)(c2-(at)2)0.5which is ugly, but there you go.
I think your initial assumption that light follows a parabola is incorrect.