A stone thrown vertically upwards with initial velocity u reaches a height ‘h’ before coming down. Show that the time taken to go up is same as the time taken to come down.

Let us consider an object which is projected vertically upwards with initial velocity  u which reaches a maximum height h.
Acceleration due to gravity=-g
Equation of Motion for a body projected thrown upwards :
V=u-gt----------(1)
h=ut-1/2gt² -----------(2)
v²-u²=-2gh --------(3)

Equations of motion for freely falling body :
for free fall : 
Initial velocity=u=0
g=g
V=gt----------(4)
h=1/2gt²-------(5)
v²=2gh------(6)
Time of Ascent is the time taken by body thrown up to reach maximum height h 
At maximum height , V=0

Equation (1) turns to u=gt1
t1=u/g  ----------(7)
Maximm height h=u²/2g ----------(8)

Time of descent : After reaching maximum height , the body begins to travel downward like free fall
so equation (5) h=1/2gt₂²
t₂²=2h/g
t₂=√2h/g
but from equation (9)
t₂=√2xu²/2g²
t₂=u/g -----------------equation (10)
∴t₁=t₂
The time ascent is equal to time of descent in case of bodies moving under gravity.