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Unformatted text preview: l\/lAE146: Astronautics w Homework #4 Assigned: Tuesday, February 37“., 2009 Due date: Tuesday , February 10mg 2009, { at the beginning; of the (:liscussirm) Please justify all of your answers and erpltuln your reasoning truth sentences. Problem 1: On gravityturn 1. m Ignoring the interaction with the atmosphere and the pressure effects, write down the equations of motion of
a rocket that maintains a constant thrust along its velocity vector (relative to the local frame) assurinng it is
launched from a site located at an altitude h, latitude c) and longitude 9. We assume the speciﬁc in’mulse of
the rocket is 13;, and its initial mass is mg. (a) First write down the equations of motion in inertial space and then in the local frame of reference (Earth
assumed to be moving uniformly about its axis passing through the poles). (b) Provide, the initial conditions in each cases. Assuming an initial mass of mg : 60,000 krnu a fuel mass of Illf : 50.000 kg, and a speciﬁc impulse of
I 5.1, 2 400 3 mass flow rate of 300 leg/s, i’nnnerically integrate the equations of motion in both frames and plot
the resulting trajectories for a launch site at Cape C ai’iaveral ( altitude of 3 m, latitude of 280 28’ and longitude
of ~80" 32’. The Earth is assumed to be spherical of mass A] 2 5.9742 X 1024 kg and radius Re : 6, 372 km).
W'l’iat is the maximum altitude reached at burnout? What is the ﬂight path angle? Problem 2: On escape velocity 1V 2, Solve Exercise #2 of Chapter 2 in your textbook, as copied below. 2. Use the energy relation to ShOW that a satellite in a hyperbolic orbit arrives at r 2: 00
with residual speed pm 2 J1 z. 25 (2.84) Soire Exercise #8 of Chapter 2 in your textbook, as copied below. 8. A spaceship consisting of a payload m and a large solar sail, as shown in Figure 2.13,
is in orbit about the sun. The sail is kept flat on to the sun and produces an acceleration from radiation pressure of
25A r cm r3 (2.88} 3,: when: S is the solar constant, A is the sail area, and (‘ is the speed of light. Show 0““ the equation of motion of the spaceship is r22‘ [1— —~ M a F 5 7 L‘ : \
ANS/X) 1‘ a) 89) ('1)? $—WWW¢£WMWKWK&4I What types of conic sections are possible when u is greater than lid/cm? When they
are equal? When [1 < BSA/cm 1’ In one of these cases the vehicle always escapes. Show
that it will arrive at r 2 :x: with residual speed ,r V ti 1/:
2 2514 l , ..
12x — I)” [1 (2.95))
\ ('m ,2 m 4  where m and M; are the initial distance and speed when the sail is unfurled. FIGURE 2.13 Solarsail spacecraft. Sun Probiem 3: On velocity expressions
1. Solve Exercise #1 of Cl’iapter 2 in your textbook, as copied ljielow. I. Show that the speed of a satellite in a circular orbit is 116. = ﬂ (2.83)
r Compare this to escape velocity at the same radius. Calculate up and um at r = 6578
km (200 km altitude) and at r :: 385,000 km (the distance to the moon). 9 Sr:>lve Exercise #4. of Chapter 2 in your textbook, as copied below. 4. One extremely confusing thing about meeting orbital mechanics for the ﬁrst time is that
almost anything can be expressed in either the geometric language of the orbital elements
or the dynamical language of energy, angular momentum, and so forth. Introduce the
ﬂightpath angle qb shown in Figure 2.12, the angle between the radius and velocity
vectors. Express the tangential and radial velocity components in terms of the energy
5, angular momentum H, the angle (15, and the radius r. FIGURE 2.12.
The ﬂightpath angle d2. Local \ horizon
\ \
\ ...
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 Winter '08
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