978-0078112829 Chapter 2 Solution Manual Part 1

Problem 2.1 Two particles of identical mass m are acted on only by the gravitational force of one upon

the other. If the distance d between the particles is constant, what is the angular velocity of the line joining

them? Use Newton’s second law with the center of mass of the system as the origin of the inertial frame.

Problem 2.2 Three particles of identical mass m are acted on only by their mutual gravitational attrac-

tion. They are located at the vertices of an equilateral triangle with sides of length d. Consider the motion

of any one of the particles about the system center of mass G and, using G as the origin of the inertial

frame, employ Newton’s second law to determine the angular velocity



required for d to remain con-

stant.

฀

Problem 2.3 Consider the two-body problem illustrated in Figure 2.1. If a force T (such as rocket

thrust) acts on

m2

in addition to the mutual force of gravitation

F

12

, show that:

(a) The equation of motion of

m2

relative to

m1

becomes:

&&

r 



r3rT

m2

.

(b) If the thrust vector T has magnitude T and is aligned with the velocity vector v, then:

TTv

v

.

v

Problem 2.4 At a given instant

t0

, an earth-orbiting satellite has the inertial position and velocity vec-

tors

r

03207ˆ

i5459ˆ

j2714 ˆ

k km

 

and

v0 6.532ˆ

i0.7835ˆ

j6.142 ˆ

kkm s

 

. Solve Equation 2.22

&&

r 



rr3

 

numerically to find maximum altitude reached by the satellite and the time at which it oc-

curs.

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 2

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Problem 2.5 At a given instant, an earth-orbiting satellite has the inertial position and velocity vectors

r

06600ˆ

i km

 

and

v012ˆ

jkm s

 

. Solve Equation 2.22

&&

r 



rr3

 

numerically to find the distance

of the spacecraft from the center of the earth and its speed 24 hours later.

Solution Using MATLAB

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Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 2

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Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 2

Problem 2.6 If r, in meters, is given by

rtsin tˆ

It2costˆ

Jt3sin 2tˆ

K

, where t is time in seconds, calcu-

late (a)

Ý

r

(where

rr

) and (b)

&

r

at

t2 s

.

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 2

Problem 2.7 Starting with Equation 2.35a (

r&

rr&

r

), prove that

&

rvˆ

ur

and interpret this result.

r&

rr&

r

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 2

Problem 2.8 Show that

ˆ

urdˆ

urdt 0

, where

ˆ

urrr

. Use only the fact that

ˆ

ur

is a unit vector. In-

terpret this result.

ur

dˆ

ˆ

dˆ

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 2

Problem 2.9 Starting with Equation 2.38

rhr3  d dt

  rr

 







, show that

ˆ

urdˆ

urdt 0

.

Solutions Manual Orbital Mechanics for Engineering Students Third Edition Chapter 2

Problem 2.10 Show that

v



h12ecos



e2

for any orbit.

Problem 2.11 Relative to a nonrotating, earth-centered Cartesian coordinate system, the position and

velocity vectors of a spacecraft are

r7000ˆ

i2000ˆ

j4000 ˆ

k km

 

and

v3ˆ

i6ˆ

j5ˆ

k km s

 

. Calculate

the orbit’s (a) eccentricity vector and (b) the true anomaly.

4053.2

Problem 2.12 Show that the eccentricity is 1 for rectilinear orbits (

h0

).

h0

e1

Problem 2.13 Relative to a nonrotating, earth-centered Cartesian coordinate system, the velocity of a

spacecraft is

v 4ˆ

i3ˆ

j5ˆ

k km s

 

and the unit vector in the direction of the radius is

ˆ

ur0.26726ˆ

i0.53452ˆ

j0.80178 ˆ

k

. Calculate (a) the radial component of velocity

vr

, (b) the azimuth

component of velocity

v

, and the flight path angle



.



 29.43