A satellite revolve around planet of radius R in an orbit of radius nR. Due to change in weather, atmospheric density increases and the satellite experiences a resistance force which depends on velocity $v$ of satellite as $f = - α v^{2} \hat{v}$ where $α$ is a positive constant. The time after which the satellite hit the surface of planet is found be $t_{0} = \frac{x m}{y α \sqrt{g R}} [\sqrt{n} - 1]$ . The value of $(x + y)$ when $x, y$ are the least possible integers is

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answer is 2.

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Detailed Solution

$k = \frac{1}{2} m v^{2} v = \sqrt{\frac{G M}{x}} d k = m v d V d k = - \frac{G M m}{2 x^{2}} d x - \frac{G M m}{x^{2}} (- d x) - α v^{2} - (V d t) = d k - \frac{G M m}{2 x^{2}} (d x) - α {(\frac{G M m}{x})}^{3 / 2} d t - \frac{m}{2} \int_{n R}^{R} \frac{d x}{\sqrt{x}} = α \sqrt{G m} \int_{0}^{t} d t t = \frac{m}{α \sqrt{g R}} (\sqrt{n} - 1)$