JEE Advanced 2023 — Paper 1 — PHY-II solution

JEE Advanced2023Paper 1PHY-II

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 $h_{Q}$ ?

A
$3 R /5$
B
$R /6$
C
$6 R /5$
D
$5 R /6$

Solution

Gravitational acceleration at orbital radius. $g \propto 1/ r^{2}$ where $r$ is distance from Earth's centre. Let $r_{P} = R + h_{P}$ and $r_{Q} = R + h_{Q}$ .

$\frac{g _{P}}{g _{Q}} = (\frac{r _{Q}}{r _{P}})^{2} = \frac{36}{25} \Rightarrow \frac{r _{Q}}{r _{P}} = \frac{6}{5}$ .

Compute $h_{Q}$ . $r_{P} = R + R /3 = 4 R /3$ , so $r_{Q} = \frac{6}{5} \cdot \frac{4 R}{3} = \frac{8 R}{5}$ .

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

Answer: (A).

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