JEE Main 2018 · Physics · Gravitation · medium

A planet of mass M and radius R has no atmosphere. A body is projected radially upward from its surface with a speed equal to the speed of a…

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A planet of mass

M

and radius

R

has no atmosphere. A body is projected radially upward from its surface with a speed equal to the speed of a satellite in a circular orbit grazing the surface. The maximum height above the surface reached by the body is

A. $R$
B. $\dfrac{R}{2}$
C. $3R$
D. $2R$

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Answer: (A) $R$

The grazing-orbit speed is

v=\sqrt{GM/R}

. Energy conservation between the surface and the highest point at distance

r

from the centre:

\tfrac12v^2-\dfrac{GM}{R}=-\dfrac{GM}{r}

. Substituting,

\dfrac{GM}{2R}-\dfrac{GM}{R}=-\dfrac{GM}{2R}=-\dfrac{GM}{r}

, so

r=2R

and the height above the surface is

r-R=R

. The option

R/2

comes from the constant-

g

formula

h=v^2/2g

, which is invalid for such large heights.

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