AEE 4263 HW 3
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AEE 4263-02 Space Flight Mechanics Name: Homework
HW3
Fall 2023
For all problems, use the following constants:
Earth gravitational parameter: μ
=
398600.4418
km
3
/
s
2
Earth radius: R
E
=
6378.137
km
1. (15 pts) Given a satellite in an elliptical Earth orbit, with semi-major axis a
=
8684.137
km
, and eccentricity e
=
0.214538
.
a)
What is the orbit’s specific angular momentum magnitude h
in km
2
/
s
?
b)
find the following parameters for true anomalies of 0, 30, 90, 120, 180, 240, 270, and 330 degrees:
i.
orbital radius r
in km
.
ii.
orbital speed v
in km
/
s
.
iii.
radial velocity component v
r
in km
/
s
.
iv.
tangential (horizontal) velocity component v
⊥
in km
/
s
.
v.
flight path angle γ
in degrees
.
c)
What are the apogee and perigee altitudes?
d)
What is the symmetry between the parameters in part b) ?
Solution:
a)
h = 57464 km^2/s
b)
i.
r(0)=6821km, r(30)=6986km, r(90)=8284km, r(120)=9280km, r(180)=10548km, r(240)=9280km, r(270)=8284km, r(330)=6986km
ii.
v(0)=8.42km/s, v(30)=8.26km/s, v(90)=7.09km/s, v(120)=6.33km/s, v(180)=5.45km/s, v(240)=6.33km/s, v(270)=7.09km/s, v(330)=8.26km/s
iii.
vr(0)=0.00km/s,
vr(30)=0.74km/s, vr(90)=1.49km/s, vr(120)=1.29km/s, vr(180)=0.00km/s, vr(240)=-1.29km/s, vr(270)=-1.49km/s, vr(330)=-0.74km/s
iv.
v
⊥
(0)=8.42km/s, v
⊥
(30)=8.23km/s, v
⊥
(90)=6.94km/s, v
⊥
(120)=6.19km/s, v
⊥
(180)=5.45km/s, v
⊥
(240)=6.19km/s, v
⊥
(270)=6.94km/s, v
⊥
(330)=8.23km/s
v.
γ
(0)=0.00deg, γ
(30)=5.17deg, γ
(90)=12.11deg, γ
(120)=11.76deg, γ
(180)=0.00deg, γ
(240)=-11.76deg, γ
(270)=-12.11deg, γ(330)=-5.17deg
c)
ha = 4169.5km, hp = 442.53km
d)
There is symmetry between 30&330deg, 90&270deg, 120&240deg, for the orbital radius, orbital
speed, and horizontal velocity. There is symmetry between 0&180deg, 30&330deg, 90&270deg, 120&240deg for the magnitude of radial velocity component and flight path but they are opposite in sign.
MATLAB:
close all clear all
clc
mu= 398600.4418; %km^3/s^2
Re=6378.137; %km
a= 8684.137; %km
e= 0.214583;
%a)
What is the orbit’s specific angular momentum magnitude
h= sqrt(mu*(1-e^2)*a)
% b) find the following parameters for true anomalies of 0, 30, 90, 120, 180,
240, 270, and 330 degrees:
% i. orbital radius r in km.
% ii. orbital speed v in km/s.
% iii. radial velocity component v_r in km/s.
% iv. tangential (horizontal) velocity component v_
⊥
in km/s.
% v. flight path angle γ in degrees.
theta= [0 30 90 120 180 240 270 330]; %deg
n=length(theta);
%i
for i=1:n
r(i)= (h^2)/(mu*(1+(e*cosd(theta(i)))));
fprintf(
'r(%g)=%0.0fkm,\t'
,theta(i),r(i))
end %ii
for i=1:n
v(i)= sqrt(mu*((2/r(i))-(1/a)));
fprintf(
'v(%g)=%0.2fkm/s,\t'
,theta(i),v(i))
end
%iii
for i=1:n
vr(i)= (mu*e*sind(theta(i)))/h;
fprintf(
'vr(%g)=%0.2fkm/s,\t'
,theta(i),vr(i))
end
%iv
for i=1:n
vperp(i)= (mu*(1+e*cosd(theta(i))))/h;
fprintf(
'v
⊥
(%g)=%0.2fkm/s,\t'
,theta(i),vperp(i))
end
%v
for i=1:n
fp(i)= atand((e*sind(theta(i))/(1+e*cosd(theta(i)))));
fprintf(
'γ(%g)=%0.2fdeg,\t'
,theta(i),fp(i))
end
%c)
What are the apogee and perigee altitudes?
ha=(h^2/(mu*(1-e)))-Re %km
hp=(h^2/(mu*(1-e)))-Re %km
%d) What is the symmetry between the parameters in part b) ?
disp(
'There is symmetry between 30&330deg, 90&270deg, 120&240deg, for the orbital'
)
disp(
'radius, orbital speed, and horizontal velocity. There is symmetry between'
)
disp(
'0&180deg, 30&330deg, 90&270deg, 120&240deg for the magnitude of radial'
)
disp(
'velocity component and flight path but they are opposite in sign.'
)
2. (5 pts) Given a spacecraft departing Earth on a hyperbolic trajectory, with a semi-major axis
a
=
10,000
km
and eccentricity e
=
1.284
, find the following:
a)
True anomaly of the asymptote θ
∞
in degrees
.
b)
Hyperbolic excess speed v
∞
in km
/
s
.
c)
Aiming radius ∆
in km
.
Solution:
a)
θ
∞
1
= 141.15deg, θ
∞
2
= 218.85deg
b)
v
∞
= 6.31km/s
c)
∆
= b = 8053.9km
MATLAB:
close all clear all
clc
mu= 398600.4418; %km^3/s^2
Re=6378.137; %km
a= 10000; %km
e= 1.284;
%a) True anomaly of the asymptote θ_∞ in degrees.
theta_inf1=acosd(-1/e) %deg
theta_inf2=360-theta_inf1 %deg
%b)
Hyperbolic excess speed v_∞ in km/s.
v_inf=sqrt(mu/a) %km/s %c) Aiming radius ∆ in km.
b= a*sqrt(e^2-1)
%km
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- Match 4 correct pairs between List-l and List-ll for questions. No credit will be given for partially correct matching. List-l A. Collision of particles B. Stability C. Satellite motion D. Spinning top List-ll 1. Euler's equation of motion 2. Minimum kinetic energy 3. Minimum potential energy 4. Impulse momentum principle 5. Conservation of moment of momentumarrow_forwardClasswork for PHYS220-A-Ali Sabbah-SP2021 Beirut Sec-A Beirut (MW 09:30-10:45) google.com/u/0/w/Mjc5NTQzMDc5NTgz/t/all PHYS220 Spring/20-21 Practice Sheet-1 Keys 1) A0.1 kg object oscillates as a s At a position X,, the object has a kinetic energy of 0.7 J and a potential energy 0.3 J. The amplitude of oscíllation, A, is: (a) 0.12 m (b) 0.22 m aple harmonic motion along the x -axis with a frequency f 3.185 Hz. (c) 0.31 m (d)0.42 m EJE thearrow_forwardHome Insert Draw View Help Class Notebook B I U Heading 1 A Dictate Times New Ro 22 QUESTION 1 A) A box full of coins weighs 105 lbs. (ignore the box weight). If the box has 65 coins and gravity on Saturn's surface is 10.44 m/s?. What is the mass and weight of coins on Saturn. Describe five physics terms you find in the problem.arrow_forward
- iz - Closes - Sunday, 16th January Midnight (page 4 of 13) - Google Chrome gmit.ie/mod/quiz/attempt.php?attempt=449467&cmid=349963&page=3 MIT From the diagram depicted, B G 55 kg C 36 kg 36 kg 5m 4m 7 m 50 kg Determine the magnitude of the weight of each traffic signal, if each signal has a mass of 36 kg, and determine the magnitude of the weight of the member OC if it has a mass of 50 kg. Express your answer in Newtons Member OC Choose. Traffic Signal Choose. age Next pagearrow_forwardF(t) Force_Crate Known values: Mass of Block 68 kg TT Hs μk 0 0.63 0.53 26° Problem Statement: A crate is initially at rest on a sloped surface, when a force is applied: F(t) where t is in seconds. Answers: When t=44.47 seconds, the crate has not yet started moving. Find the magnitude of the friction force at that time: 317.771 N The direction of the friction force at that time is: down the slope Find the time required for the crate to start moving: Find the crate's speed when t=174.9 s: 9.485 m/s Be sure to include units with your answers. 169.756 S 240+0.2 N,arrow_forwardPHYS X PHYS X 印 PHYS X PHYS X POTPHYS X PHYS X E PHYS X E PHYS top/semester2/physics%20for%20engineers/PHYS220_CH15_Lecture%20Notes_Problems%2015 19,15.29 S (D Page view A Read aloud V Draw Problem-15-19: page-475 A 0.500-kg object attached to a spring with a force constant of 8.00 N/m vibrates in simple harmonic motion with an amplitude of 10.0 cm. Calculate the maximum value of its (a) speed, and acceleration. (b) the speed and the acceleration when the object is 6.00 em from the equilibrium position, and (c) the time interval required for the object to move from.r50 to r5 8.O0 cm. Solution:arrow_forward
- Remaining Time: 46 minutes, 11 seconds. * Question Completion Status: A Moving to another question will save this response. Question 4 A bob mass in a conical pendulum attached by a cord makes an angle 0= 10° with the vertical and moves with a constant speed. Find the centripetal acceleration (in m/s-) of the bob mass. O 5.77 O 2.68 O 4.66 O 1.76 O 3.64 A Moving to another question will save this response.arrow_forwardFTA FTB Two blocks are connected by a light string passing over a pulley of radius 0.18 m and moment of inertia I. The blocks move (towards the right) with an acceleration of 1.4 m/s² along their frictionless inclines (see). mA = 8.0 kg 32° Copyright © 2008 Pearson Education, Inc. a 61% mB = 10.0 kgarrow_forwardProblem Set 1- VECTOR RESULTANT BY ANALYTICAL METHOD PROBLEM SOLVING: Solve the following problems. Show pertinent solution and box the final answer. 1. The figure below represents a cart viewed from above being pulled by two children and experiencing a frictional force. Determine the net force on the sled by graphical method using a scale factor 1.0 cm = 4.0 N. F, = 8 N 30 "45 Cart f = 6 N F, = 10 Narrow_forward
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