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A satellite is in an elliptic orbit around the earth with aphelion of 6R and perihelion of 2R where R = 6400 km is the radius of the earth . Find the velcoity of the satellite at apogee and perigee.[ G = 6.67 xx10^(-1) SI unit and M= 6 xx 10^(24)kg ]

Answer» <html><body><p></p>Solution :`implies` Given , `r_p` = radius of perihelion = 2R <br/>`r_a` = radius of aphelion = 6R <br/> Hence , we can write, <br/> `r_(a) = a(1+e) = 6R "" ...(i)` <br/> `r_p = a (1-e) = 2 R ""...(ii)` <br/> <a href="https://interviewquestions.tuteehub.com/tag/solving-1217196" style="font-weight:bold;" target="_blank" title="Click to know more about SOLVING">SOLVING</a> equ (i) and (ii) ,we get <br/> <a href="https://interviewquestions.tuteehub.com/tag/eccentricity-964854" style="font-weight:bold;" target="_blank" title="Click to know more about ECCENTRICITY">ECCENTRICITY</a> , `e =1/2` <br/> Angular <a href="https://interviewquestions.tuteehub.com/tag/momentum-1100588" style="font-weight:bold;" target="_blank" title="Click to know more about MOMENTUM">MOMENTUM</a> remains unchanged <br/> `:. mv_pr_p = mv_ar_a` <br/> `:. (v_a)/(v_p) = r_p/r_a = (2R)/(6R) =1/3` <br/> Energy is same at perigee and apogee, <br/> `1/2mv_p^2-(GMm)/r_p=1/2mv_a^2-(GMm)/r_a` <br/>`:. v_p^2(1-1/9)=-2GM(1/r_a-1/r_p)=2GM(1/r_p-1/r_a)` <br/> (By putting `v_a=v_p/3`) <br/> `v_p=([2GM(1/r_p-1/r_a)]^(1/2))/([1-(v_a/v_p)^2]^(1/2))=[((2GM)/R(1/2-1/6))/((1-1/9))]^(1/2)` <br/> `=((3/2GM)/(8/9R))^(1/2)=sqrt(3/4(<a href="https://interviewquestions.tuteehub.com/tag/gm-1008640" style="font-weight:bold;" target="_blank" title="Click to know more about GM">GM</a>)/R) ` = 6.85 km/s<br/> `v_p = 6.85km//s` <br/> For circular orbit of radius r, <br/> `v_c` = orbita velocity = `sqrt((GM)/r)` <br/> For `r = 6R ,v_c =sqrt((GM)/(6R)) = 3.23 km//s` <br/> Hence , to increase to a circular orbit at apogee, we have to increase the velocity by (3.22-2.28) = 0.95 km/s . This can be done by suitably firing rockets from the <a href="https://interviewquestions.tuteehub.com/tag/satellite-1195060" style="font-weight:bold;" target="_blank" title="Click to know more about SATELLITE">SATELLITE</a> .</body></html>


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