A \(100 \mathrm{~kg}\) spacecraft is in circular orbit around the earth, with orbital radius \(10^{4} \mathrm{~km}\) and with speed \(6.32 \mathrm{~km} / \mathrm{s}\). It is desired to turn on the rocket engines to accelerate the spacecraft up to a speed so that it will escape the earth and coast out to Jupiter. Use a value of \(1.5 \times 10^{8} \mathrm{~km}\) for the radius of earth's orbit, \(7.8 \times 10^{8} \mathrm{~km}\) for Jupiter's orbital radius, and a value of \(30 \mathrm{~km} / \mathrm{s}\) for the velocity of the earth. Determine

(a) the semi-major axis of the Hohmann transfer orbit to Jupiter;

(b) the travel time to Jupiter;

(c) the heliocentric velocity of the spacecraft as it leaves the earth;

(d) the minimum \(\Delta v\) required from the engines to inject the spacecraft into the transfer orbit.