The ball will reach maximum height (h) when velocity of ball is zero.

Using, ( 0 )² - u² = 2 ( -g ) h

So,  h  = u² / 2 g = ( 100 )² / 2 g

So, velocity of ball at half the max. height ( h / 2 ) will be given by,

v² - u² = 2 ( - g ) ( h / 2 )

or, v² = ( 100 )² - ( 100 )² / 2

or, v = 100 ( 1 / 2 )^1/2 = 50 ( 2 )^1/2   (Ans)............