Question

prove that for an earth satellite,the ratio of the velocity at apogee to its velocity at perigee is equal to the inverse ratio of its distances from apogee and perigee​

Answer 1

Dear Student ,
Let us consider that the v_a and r_a be the velocity and the distance of apogee position and v_b and r_b are the velocity and distance of the perigee position of the satellite of earth .
Now as the angular momentum is conserved so we can write that ,
$$\begin{gathered} L_a=L_p=cons\tan t \\ \Rightarrow m{v}_a{r}_a=m{v}_p{r}_p\left[m=massofthesatellite\right] \\ \Rightarrow \frac{v_a}{v_p}=\frac{r_p}{r_a}\left[Proved\right] \end{gathered}$$
Regards