Question

A particle is projected vertically upward from surface of earth with a kinetic energy equal to half the minimum value needed for it to escape

Answer 1

Hey mate ^_^

Here is ua answer.... ⏬⏬

Check this attachment :-)

Answer 2

"The kinetic energy which is to escape from the surface is given as,

                $E\quad =\quad \frac { 1 }{ 2 } m{ v }^{ 2 }\\=\quad \frac { 1 }{ 2 } m(\sqrt { \frac { 2Gm }{ r } } )^{ 2 }\\=\quad \frac { 1 }{ 2 } m(\frac { 2Gm }{ r } )\\=\quad \frac { Gm }{ r }$

Potential energy at the surface of the earth+ ½ E =Potential energy at the maximum height

                  $\frac { -GMm }{ r } +\frac { GMm }{ 2r } \quad =\quad \frac { -GMm }{ r+h }\\\frac { -GMm }{ 2r } \quad =\quad \frac { -GMm }{ r+h }\\\frac { 1 }{ 2r } \quad =\quad \frac { 1 }{ r+h }\\2r =r+h\\h = r$