If the body travels half its total path in the last second of its fall from rest, find:
(a) The time and (b) height of its fall. Explain the physically unacceptable solution of the quadratic time equation. `(g=9.8 m//s^(2))`

Explanation

If the body falls a height h in time t, then `" "h=1/2 g t^(2)`
`[u=0 " as the body starts from rest"]` …(i)
Now, from Equation (i) and (ii) distance travelled in the last second.
`h-h'=1/2g t^(2)-1/2g(t-1)^(2)` i.e., `h-h'=1/2 g(2t-1)`
But according to given problem as `(h-h')=h/2`
`rArr (1/2)h=(1/2)g(2t-1)` or `(1/2)g t^(2)=g(2t-1)` [as from equation (i) `h=(1/2)g t^(2)`]
`rArr t^(2)-4t+2=0" "` or `" " t=[4+-sqrt((4^(2)-4xx2)"]"//2)rArr t,=2+-sqrt(2)rArr t=0.59s` or `" "3.41 s`
0.59 s is physically unacceptable as it gives the total time t taken by the body to reach ground lesser than one sec while according to the given problem time of motion must be greater than `1s`.
so `t=3.41 s` and `h=1/2xx(9.8)xx(3.41)^(2)=57 m`