First slide

Motion of planets and satellites

Question

A body A of mass m is moving in a circular orbit of radius R about a planet . Another body B of mass $\frac{m}{2}$ collides with A with a velocity which is half $(\frac{\vec{v}}{2})$ the instantaneous velocity $\vec{v}$ of A. The collision is completely inelastic . Then the combine body:

Moderate

A

Falls vertically downwards towards the planet
B

Escapes from the Planet’s Gravitational field
C

Starts moving in an elliptical orbit around the planet
D

continuous to move in a circular orbit

Solution

$\begin{array}{l} L e t s a y i n i t i a l d i r e c t i o n o f v e l o c i t y o f A a l o n g \tan g e n t i s i n y a x i s \\ a n d c o l l i s i o n i n h e a d o n . \\ A p p l y i n g M o m e n t u m C o n s e r v a t i o n, \\ {\vec{p}}_{i} = {\vec{p}}_{f} \\ \Rightarrow m v ({\hat{j}}^{}) - \frac{m}{2} . \frac{v}{2} ({\hat{j}}^{}) = (m + \frac{m}{2} ) {\vec{v}}_{f} \\ \Rightarrow v ({\hat{j}}^{}) - \frac{v}{4} ({\hat{j}}^{}) = (\frac{3}{2} ) {\vec{v}}_{f} \\ \Rightarrow \frac{3 v}{4} ({\hat{j}}^{}) = (\frac{3}{2} ) {\vec{v}}_{f} \\ \Rightarrow {\vec{v}}_{f} = \frac{v}{2} \hat{j} \\ A f t e r c o l l i s i o n, t h e f i n a l s p e e d i s n o t e q u a l t o o r b i t a l s p e e d a t t h a t p o i n t . S o m o t i o n c a n n o t b e c i r c u l a r . \\ S i n c e v e l o c i t y w i l l r e m a i n \tan g e n t i a l, s o i t c a n n o t f a l l v e r t i c a l l y t o w a r d s t h e p l a n e t . \\ S p e e d a f t e r c o l l i s i o n i s l e s s t h a n e s c a p e s p e e d \sqrt{2} v, s o t h e y c a n n o t e s c a p e g r a v i t a t i o n a l f i e l d . \\ S o t h e i r m o t i o n w i l l b e e l l i p t i c a l a r o u n d t h e p l a n e t . \end{array}$

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