both working for method 1 and method 2 are a bit ambiguous because you have not stated clearly your u & v are velocities at which points?

note that the object is accelerating as it falls so velocities at gates 1,2 & 3 are all different.

(s1)/t actually gives the average velocity between gate 1 & 2, and and (s2)/t gives the average velocity between gate 2 & 3.

i think your method 2 did not work because you are using the square of the average velocities. i have not tried, but i think if you consider the velocities at the 3 gates separately, it is possible that the method 2 may still work.

alternatively, a shorter & faster working is ...
let velocity of object at gate 2 be V, then
s1 = Vt-(1/2)at^2 ...... (1)
s2 = Vt+(1/2)at^2 ...... (2)
and (2)-(1) gives you a=(s2-s1)/(t^2).

To handle questions on kinematics, we can pick the appropriate formula to be used. in this case s = ut + 1/2 a t^2 for s1 and for s2
We then use eqn 2 - eqn 1 and then make a the subject to get the answer.
Your main problem is using v = s/2 wrongly. This is average velocity but not instantaneous velociy. (Make sure you understand the difference)