P206 7.1 Problem 1: A satellite is placed into a circular orbit by launching it

Question

P206 7.1

Problem 1: A satellite is placed into a circular orbit by launching it with a two-stage rocket. The first stage provided the rocket with speed vo inclined from the vertical by and elevation angle e-on reaching of the initial orbit, the second stage is ignited, generating a boost v1 that places the payload into a circular orbit. Let Vo-8 km/s, and °-35°. (Ignore air resistance and the rotational motion of the Earth) a) Calculate the additional speed boost 1, required of the second stage to make the final orbit circular. b) Calculate the altitude h of the final orbit. Note: As usual there are different ways to correctly solve the problem, and in some methods one could solve part (b) before part (a), and that is completely fine o i Earth v,+Av NOT TO SCALE

Explanation / Answer

given

initial speed, vo = 8000 m/s

thetao = 35 deg

M = mass of earth = 5.972*10^24 kg

Re = radius of earth = 6371,000 m

so the final orbit is at a distance r1, at this distance of the intiial orbit the velocity boost dv is provided

a. form conservation of angular momentum, for satellite of mass m

mvo*sin(theta0)Re = m*v1*r1

Re = radius of earth = 6371,000 m

also, from conservation of energy

0.5mvo^2 - GMm/Re = 0.5mv1^2 - GMm/r1

8000*sin(35)*6371,000 = v1*r1 = 29234043807.9401174

and

vo^2 - 2GM/Re = v1^2 - 2GM*v1/29234043807.9401174

v1^2 - 27251.26928v1 + 61045487.364 = 0

v1 = 2462.64 m/s

and r1 = 11,871,014.2695 m

now, for a circular orbit at radius r1

speed required = v

GM/r1 = v^2

v = 5792.6715 m/s

hence

dv = v - v1 = 3330.0315 m/s

b. h = r1 - R = 5500014.2695 m

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