Solution Manual for Orbital Mechanics for Engineering Students, 4th Edition by Howard D. Curtis

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SOLUTIONS MANUAL

to accompany


ORBITAL MECHANICS FOR ENGINEERING STUDENTS




Howard D. Curtis
Embry-Riddle Aeronautical University
Daytona Beach, Florida




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Solutions Manual Orbital Mechanics for Engineering Students Chapter 1


Problem 1.1
(a)
( )(
A ⋅ A = Ax iˆ + Ay ˆj + Az kˆ ⋅ Ax iˆ + Ay ˆj + Az kˆ )
( ) ( )
= Ax iˆ ⋅ Ax iˆ + Ay ˆj + Az kˆ + Ay ˆj ⋅ Ax iˆ + Ay ˆj + Az kˆ + Az kˆ ⋅ Ax iˆ + Ay ˆj + Az kˆ ( )
( ) ( ) ( )
=  Ax 2 ( iˆ ⋅ iˆ ) + Ax Ay iˆ ⋅ ˆj + Ax Az ( iˆ ⋅ kˆ )  +  Ay Ax ˆj ⋅ iˆ + Ay 2 ˆj ⋅ ˆj + Ay Az ˆj ⋅ kˆ 
   
( )

( )
+  Az Ax ( kˆ ⋅ iˆ ) + Az Ay kˆ ⋅ ˆj + Az 2 ( kˆ ⋅ kˆ ) 

=  Ax (1) + Ax Ay ( 0 ) + Ax Az ( 0 ) +  Ay Ax ( 0 ) + Ay 2 (1) + Ay Az ( 0 ) +  Az Ax ( 0 ) + Az Ay ( 0 ) + Az 2 (1)
2
     
= Ax 2 + Ay 2 + Az 2


But, according to the Pythagorean Theorem, Ax2 + Ay 2 + Az 2 = A2 , where A = A , the magnitude of
the vector A . Thus A ⋅ A = A2 .

(b)
iˆ ˆj kˆ
A ⋅ ( B × C ) = A ⋅ Bx By Bz
Cx Cy Cz

(  ) (
= Ax iˆ + Ay ˆj + Az kˆ ⋅  iˆ By Cz − Bz C y − ˆj ( Bx Cz − Bz Cx ) + kˆ Bx C y − By Cx 
)  ( )
( )
= Ax By Cz − Bz C y − Ay ( Bx Cz − Bz Cx ) + Az Bx C y − By Cx ( )
or

A ⋅ ( B × C) = AxBy C z + Ay Bz C x + Az BxC y − AxBz C y − Ay BxC z − Az By C x (1)

Note that ( A × B) ⋅ C = C ⋅ ( A × B) , and according to (1)

C ⋅ ( A × B) = C x Ay Bz + C y Az Bx + C z AxBy − C x Az By − C y AxBz − C z Ay Bx (2)

The right hand sides of (1) and (2) are identical. Hence A ⋅ ( B × C) = ( A × B) ⋅ C .

(c)
iˆ ˆj kˆ iˆ ˆj kˆ
(
A × ( B × C ) = Ax iˆ + Ay ˆj + Az kˆ × Bx ) By Bz = Ax Ay Az
Cx Cy Cz By Cz − Bz C y Bz Cx − Bx C y Bx C y − By Cx

=  Ay Bx C y − By Cx − Az ( Bz Cx − Bx Cz )  î +  Az By Cz − Bz C y − Ax Bx C y − By Cx  ˆj
( ) ( ) ( )
   
)
+  Ax ( Bz Cx − Bx Cz ) − Ay By Cz − Bz C y  k̂
  (
= Ay Bx C y + Az Bx Cz − Ay By Cx − Az Bz Cx iˆ + Ax By Cx + Az By Cz − Ax Bx C y − Az Bz C y ˆj
( ) ( )
+ ( Ax Bz Cx + Ay Bz C y − Ax Bx Cz − Ay By Cz ) k̂

= Bx ( Ay C y + Az Cz ) − Cx ( Ay By + Az Bz )  î + By ( Ax Cx + Az Cz ) − C y ( Ax Bx + Az Bz )  ĵ
   
+ Bz ( Ax Cx + Ay C y ) − Cz ( Ax Bx + Ay By )  k̂
 

Add and subtract the underlined terms to get




1


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Solutions Manual Orbital Mechanics for Engineering Students Chapter 1



( ) (
A × ( B × C ) = Bx Ay C y + Az Cz + Ax Cx − Cx Ay By + Az Bz + Ax Bx  î
  )
( ) (
+ By Ax Cx + Az Cz + Ay C y − C y Ax Bx + Az Bz + Ay By  ĵ
  )
+ Bz Ax Cx + Ay C y + Az Cz − Cz Ax Bx + Ay By + Az Bz  kˆ
( ) ( )
 
( ˆ ˆ ˆ
)( ˆ
) ( ˆ
)(
= Bx i + By j + Bz k Ax Cx + Ay C y + Az Cz − Cx i + C y j + Cz kˆ Ax Bx + Ay By + Az Bz )
or

A × ( B × C) = B( A ⋅ C) − C( A ⋅ B)

Problem 1.2 Using the interchange of Dot and Cross we get

( A × B) ⋅ (C × D) = [( A × B) × C] ⋅ D
But

[( A × B) × C] ⋅ D = − [C × ( A × B)] ⋅ D (1)

Using the bac – cab rule on the right, yields

[( A × B) × C] ⋅ D = −[ A(C ⋅ B) − B(C ⋅ A)] ⋅ D
or

[( A × B) × C] ⋅ D = −( A ⋅ D)(C ⋅ B) + ( B ⋅ D)(C ⋅ A) (2)

Substituting (2) into (1) we get

[( A × B) × C] ⋅ D = ( A ⋅ C)( B ⋅ D) − ( A ⋅ D)( B ⋅ C)
Problem 1.3
Velocity analysis

From Equation 1.38,

v = v o + Ω × rrel + v rel . (1)

From the given information we have

v o = −10Iˆ + 30 Jˆ − 50K
ˆ (2)


(
rrel = r − ro = 150Iˆ − 200 Jˆ + 300K ) (
ˆ − 300Iˆ + 200 Jˆ + 100K )
ˆ = −150Iˆ − 400 Jˆ + 200K
ˆ (3)


Iˆ Jˆ ˆ
K
Ω × rrel = 0.6 −0.4 1.0 = 320Iˆ − 270
0 Jˆ − 300K
ˆ (4)
−150 −400 200




2


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