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Question:

India's Mangalyan was sent to the Mars by launching it into a transfer orbit EOM around the sun. It leaves the earth at E and meets Mars at M. If the semi-major axis of Earth's orbit is ae=1.5×10¹¹m, that of Mars's orbit am=2.28×10¹¹m, taken Kepler's laws give the estimate of time of Mangalyan to reach Mars from Earth to be close to : 500days, 320days, 260days, 220days

320days

260days

500days

220days

Solution:

Consider the trajectory to be an Ellipse,Semi-major axis,a=ae+am/2=1.89×10¹¹mRadius of the earth =1.5×10¹¹mNow,We consider a circle whose area is equal to the path covered by ellipse,∴πR²=πab/2∴R=√ab/2=√1.89×10¹¹×1.5×10¹¹/2=1.19×10¹¹mApplying kepler's law,∴T²/r³=T²/R³∴(T/365)²=(Rae)³∴T²=(1.19/1.5)³×3.65²∴T≈260days.