A body is projected vertically upwards from the surface of a planet of radius `R` with a velocity equal to half the escape velocity for that planet. The maximum height attained by the body is

Options

A. R/2
B. R/3
C. R/5
D. R/4

Answer

B

Explanation

Escape velocity`=v_(e)=sqrt((2GM)/(R))`
where M and R are the mass and radius of the planet respectively.
The initial kinetic energy of the body`=(1)/2 mv^(2)`
`=(1)/(2)m((v_(e))/(2))^(2)=(1)/(2)m(2GM)/(4R)=(GMm)/(4R)`
Initial potential energy on the surface of the planet
`=-(GMm)/(R)`
`therefore` Total enegy`=(GM m)/(4R)-(GM m)/(R)=-(3)/(4)(GM m)/(R)`
Let the maximum height attained by the body be h. the kinetic enegy at that point is zero.
Potential energy at that height`=-(GMm)/(R+h)`
`therefore` Total energy`=0-(GM m)/(R+h)=-(GM m)/(R+h)`
`therefore` From the principle of convervation of energy
`-(GMm)/(R+h)=-(3)/(4)(GM m)/(R)`
`implies3R+3h=4Rimpliesh=R//3`.