\"Flight to Mars. To send a satellite from Earth to Mars, a rocket must accelera
ID: 2062425 • Letter: #
Question
"Flight to Mars. To send a satellite from Earth to Mars, a rocket must acceleratethe satellite until it is in the correct elliptical orbit around the sun. The satellite does
not travel to Mars under rocket power, because that would require more fuel than
it could carry. It just moves on a Keplerian orbit under the influence of the sun’s
gravity.
The satellite orbit must have perihelion distance r- = RE (= radius of Earth’s
orbit) and aphelion distance r+ = RM (= radius of Mars’s orbit) as shown in the
figure. The planetary orbit radii are
RE = 1.496E11 m and RM = 2.280E11 m.
Calculate the time for the satellite’s journey"
Explanation / Answer
Two set of topics to be known
1) Newton's second law says F = ma.
The centripetal acceleration in a = v2=R where R is the radius of the orbit and v is the speed (=2R/T ).
2)The gravitational force is F = GMm=R2.
Combine these equations 1 and 2
[m(2R/T )^2]/R=GMm/R^2 ;
from the above T=sqrt(4^2R^3/GM)
but when we consider an elliptic path , here R= semi major axis that is distance between sun and mars.
R=1.523679 AU = 2.279423*10^11
G=6.673*10^-11 m^3 kg^-1 s^-2
Mass of sun, M=1.98892*10^30 kg
neglecting radius of mars and earth compared to distance from sun and neglecting mass of satellite compared to mass of sun
therfore T=59353732.85 sec = 686.9645 days.
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