e) Show that the orbital:speed of the earth is \( v=\sqrt{\frac{G M}{r}} \), Where \( r \) the sepatation between the earth and sun, \( G \) is the universal: gravitational constant, and M, is mass of the sun. [3]
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The gravitational force \( F \) is given by: \[ F = \frac{G M m}{r^2} \] where \( m \) is the mass of the Earth, \( M \) is the mass of the Sun, \( r \) is the separation between the Earth and the Sun, and \( G \) is the universal gravitational constant. Show more…
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4. Show that for a satellite of mass ( m ) moving with velocity ( v ) in a circular orbit of radius ( r ) about an attracting center of mass ( M ),[v=sqrt{frac{G M}{r}}=frac{v_{e}}{sqrt{2}}, quad l=m sqrt{G M r}]where ( v_{0} ) is the escape velocity, and ( l ) is the orbital angular momentum.
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Show that the velocity of escape from the Sun at the Earth's distance from the Sun is $\sqrt{2}$ times the speed of the Earth in its orbit, assumed to be a circle. (This is a specific case of a general result for circular orbits: $v_{\text {esc }}=\sqrt{2} v_{\text {orb }} .$ )
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