N12R.3Imagine you are the orbital engineer for the first NASAspace shot to Ceres, the largest known asteroid. Ceres’s nearlycircular orbit around the sun has a radius of R = 2.77 AU. Afterbeing launched from the earth, the probe will initially be in acircular orbit around the sun with the same radius (rc = 1.0 AU)and the same orbital speed Inline_Formula as the earth’s orbit. Theprobe’s rocket engines will then fire briefly to increase theprobe’s speed to that speed Inline_Formula needed to put the probeinto an elliptical orbit whose initial (and minimum) distance fromthe sun is rc = 1.0 AU and whose final (and largest) distance fromthe sun is R = 2.77 AU (this is a Hohmann transfer orbit). (a) Howlong will it take the probe to get from the earth to Ceres in suchan orbit? (b) What is the speed Inline_Formulathat the probe has tohave just after firing its engines to be inserted into this orbit?(Assume the duration of the boost is short enough that its distancefrom the sun is still nearly rc just after the engines have beenfired.)
