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d. 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.

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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

V=gt----------(4)

h=\(1\over2\)gt²-------(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

\(t_1=\tfrac{u}{g} \) ----------(7)

Maximm height h=\(u²\over2g\) ----------(8)

Time of descent : After reaching maximum height , the body begins to travel downward like free fall

so equation (5) h=\(1\over2\)gt₂²

t₂²=\(2h\over g\)

\(t₂=\sqrt{\tfrac{2h}{g}}\)

but from equation (9)

t₂=\( \sqrt{\tfrac{2xu²}{2g²}}\)

t₂=u/g -----------------equation (10)

The time ascent is equal to time of descent in case of bodies moving under gravity

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