Let x1 be the total distance particle falls. Let t1 be the time taken to fall this distance. Then:

u = 0

a = g

t = t1

s = x

Substitute all this into an appropriate straight line motion formula to get equation 1.

If it falls 5/9 of total fall in last second, then it falls 4/9 of total fall in t1 - 1 seconds:

u = 0

a = g

t = t1 - 1

s = (4/9) x

Substitute all this into an appropriate straight line motion formula to get equation 2.

Solve equations (1) and (2) simultaneously for x and t1.