Two satellites P and Q are moving in different circular orbits around the Earth (radius $R$). The heights of P and Q from the Earth surface are $h_{\mathrm{P}}$ and $h_{\mathrm{Q}}$, respectively, where $h_{\mathrm{p}}=\mathrm{R} / 3$. The accelerations of P and Q due to Earth's gravity are $g_{\mathrm{P}}$ and $g_{\mathrm{Q}}$, respectively. If $g_{\mathrm{P}} / g_{\mathrm{Q}}=36 / 25$, what is the value of $h_{\mathrm{Q}}$?

JEE Advanced · 2023 · 04 Jun 2023 · Paper 1 · Gravitation

Two satellites P and Q are moving in different circular orbits around the Earth (radius

R

). The heights of P and Q from the Earth surface are

h_{P}

and

h_{Q}

, respectively, where

h_{p} = R /3

. The accelerations of P and Q due to Earth's gravity are

g_{P}

and

g_{Q}

, respectively. If

g_{P} / g_{Q} = 36/25

, what is the value of

h_{Q}

Answer: (A)

The gravitational acceleration at distance

r

from Earth's center is

g = \frac{GM}{r ^{2}}

.

For satellite P:

r_{P} = R + h_{P} = R + \frac{R}{3} = \frac{4 R}{3}

For satellite Q:

r_{Q} = R + h_{Q}

\frac{g _{P}}{g _{Q}} = \frac{r _{Q}^{2}}{r _{P}^{2}}

\frac{36}{25} = \frac{r _{Q}^{2}}{( \frac{4 R}{3} ) ^{2}}

\frac{r _{Q}}{r _{P}} = \frac{6}{5}

r_{Q} = \frac{6}{5} \times \frac{4 R}{3} = \frac{8 R}{5}

h_{Q} = r_{Q} - R = \frac{8 R}{5} - R = \frac{3 R}{5}

JEE Advanced · 2023 · 04 Jun 2023 · Paper 1 · Single correct option