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  4. A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be: (Take radius of earth =6400 km and g =10 ms -2 )

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Q. A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be: (Take radius of earth $=6400 \,km$ and $g =10\, ms ^{-2}$ )

JEE MainJEE Main 2022Gravitation

Solution:

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$v _{ e }=\sqrt{\frac{2 Gm }{ R }}$
$ \frac{- GMm }{ R }+\frac{1}{2} m \frac{ V _{ e }^2}{9}=-\frac{ GMm }{ R + h } $
$ \frac{ GM }{ R + h }=\frac{ GM }{ R }-\frac{ V _{ e }^2}{18}$
$ \frac{ GM }{ R + h }=\frac{ GM }{ R }-\frac{ GM }{9 R }$
$ \frac{ GM }{ R + h }=\frac{8 GM }{9 R } $
$ \frac{1}{ R + h }=\frac{8}{9 R } $
$ 9 R =8 R +8 h$
$ h =\frac{ R }{8} \Rightarrow \frac{6400}{8} \Rightarrow 800\, km $