question_answer 55)

                  Show the nature of the following graph for a satellite orbiting die earth.                 (a) KE \[\upsilon s\] orbital radius R                 (b) PE \[\upsilon s\] orbital radius R                 (c) TE \[\upsilon s\] orbital radius R.       

Answer:

                  Since \[K=\frac{GMm}{2r}=|E|=\frac{|U|}{2},\] all energies vary with r hyperbolically. Hence, \[(K\,vs\,r),\,(U\,vs\,r)\]and \[(E\,vs\,r)\] graphs are rectangular hyperbolas.                 Negative total energy \[\left( E=\frac{GMm}{2r} \right)\] signifies that the system [Earth (M) + Satellite (m)] is a bounded system.                                                 If the satellite has an orbit of radius r, its total energy must be negative.                 If \[E\ge U,\] the satellite will no more be bound to the Earth.                 When \[r\to 0,\,|Energy|\to \infty \] and when \[r\to \infty ,\,|Energy|\to 0.\]