A satellite is transferred from one circular orbit to another circular orbit by Hohmann Transfer orbit maneuver, by temporarily entering in an intermediate elliptical orbit by increasing speed instantly by firing propeller. To transfer a satellite in circular orbit of radius $r$ to another orbit of radius $3 r$ , additional energy to be imparted at point P (in orbit of radius r) is $n \frac{G M}{r}$ .Find $n$ .( $M$ = Mass of planet)

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Detailed Solution

Let speed of the satellite at P and Q be V and V’ respectively.

By conservation of angular momentum, between Q and P, we get

$\Rightarrow m V' (3 r) = m v r \Rightarrow V' = \frac{V}{3} \dots \dots \dots \dots (i)$

By conservation of energy, between Q and P, we get

$\Rightarrow \frac{1}{2} m {v^{'}}^{2} - \frac{G M m}{3 r} = \frac{1}{2} m v^{2} - \frac{G M m}{r} \Rightarrow \frac{1}{2} m {(v)}^{2} - \frac{1}{2} m {(\frac{v}{3})}^{2} = \frac{G M m}{r} - \frac{G M m}{3 r} \frac{3 G M m}{4 r} - \frac{G M m}{2 r} = Δ K ∴ Δ k = \frac{1}{4} \frac{G M m}{r}$