The New Horizons spacecraft, launched in 2006, approached the quasi-planet Pluto at an altitude of 14,000 km with an orbital velocity of 14 km/s in July 2015. The spacecraft then performed a “flyby,” passing by Pluto without entering its orbit.

Suppose that the spacecraft did not perform a flyby, but instead, as shown in the Figure, the spacecraft thrusted its propellant back at point A on orbit I and entered an elliptical orbit (orbit II) with an eccentricity e = 0.8 and apogee altitude H = 14,000 km. Answer the following questions for this case. Assume that Pluto’s radius *Ro* is 1185 km, the gravitational parameter *μp* is 8.749 x 10^2 (km^3/s^2), and the velocity V on orbit I is 14 km/s. Calculate to five significant digits.

(1) Find the semi-major axis a, semi-minor axis b, and semi-focal distance c of the elliptical orbit (orbit II).

(2) Find the orbital velocity *Va* at the apogee of the elliptical orbit (orbit II), and find the difference in velocity ΔV between before and after the injection.

(3) At perigee B on the elliptical orbit (orbit II), a reverse thrust is made to shift to a circular orbit (orbit III) with e=0. Find the orbital altitude h and orbital velocity *Vb* .

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