# A ballon moves up vertically such that if a stone is thrown from it with a horizontal velocity v_(0) relative to it the stone always hits the ground

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A ballon moves up vertically such that if a stone is thrown from it with a horizontal velocity v_(0) relative to it the stone always hits the ground at a fixed point 2v_(0)^(2)//g horizontally away from it. Find the height of the balloon as a function of time.

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Range =v_(0)xx"time of flight" (t)
rArrt=(2v_(0)^(2))/(gv_(0))=(2v_(0))/(g)
If y is the height of balloon at any instant t and (dy)/(dt) its velocity then
y= -((dy)/(dt))((2v_(0))/(g))+(1)/(2)g((2v_(0))/(g))^(2)
 rArr (dy)/(dt)+(g)/(2v_(0))y-v_(0)=0
(dy)/(v_(0)-(g)/(2v_(0))y)=dt
 rArr -(2v_(0))/(g)ln(v_(0)-(g)/(2v_(0))y)=t+c
At t=0, y=0rArrc= -(2v_(0))/(g)lnv_(0)
Simplfying we get
y=(2v_(0)^(2))/(g)[1-e^(-"gt"//2v_(0))]