The mass of the Sun M = 2.0×1030 kg, and G = 6.67×10-11 Nm2 /kg2 a)....

The mass of the Sun M = 2.0×1030 kg, and G = 6.67×10-11 Nm2 /kg2

a). A spaceship of mass m = 7.5×104 kg is on a circular orbit of radius r1 = 2.5×1011 m around the Sun. The captain of that spaceship decides to increase the radius of his orbit to r2 = 4.0×1011 m. What is the minimum amount of energy he has to expand using his engines to move to this higher orbit? [Assume that the ship has constant mass when changing orbits; perhaps it has an atomic engine. You need to find ΔE = E2 – E1, where E is the total energy on an orbit (kinetic plus potential). Derive the formula for ΔE expressed through known quantities m, M, r1, r2 and G, and, at the end, substitute the numbers]

b). After some maneuvering, the spaceship momentarily stops at a distance r = 3.0×1011 m from the Sun [When it stops, it is no longer on an orbit; at that moment it has only negative potential energy with respect to the Sun and no kinetic energy]. Then the captain fires his engines for a short time to give his ship initial velocity Vesc so that the ship would “escape” from the Solar system (he is fed-up with the Solar system and he just wants to go far away from it). From a statement of conservation of energy, derive a formula and then calculate the value of that initial velocity Vesc

c). Now the ship is far away from the Solar system, and it essentially ceased moving with respect to the Sun. So it “escaped”, but it is still far away from its home star Alpha Centauri. The captain now decides to cruse to his home star with cruising velocity v = 80 km/s. He points his ship towards his home star and fires his engines. What minimum amount of fuel energy he has to use to acquire that cruising velocity?

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