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MAE 4410: Fundamentals of Astronautics
Homework #1
1. Eratosthenes (276-196 BC) was one of the first ancient astronomers to calculate
the size of the Earth. He made a very simple model for this that involved the
following assumptions:
1. The Earth is a sphere.
2. The Sun is very far away.
Eratosthenes knew that at noon on June 21 (the summer solstice - longest day),
the Sun would shine into the bottom of a deep vertical pit in the city of Syene,
meaning the Sun was at the zenith in the sky at this time. Being in Alexandria,
which was located more or less due north of Syene, he could measure the length
of a shadow cast by an obelisk at the same time, noon on the summer solstice.
This measurement, along with the height of the obelisk, gave him the angle that
the Sun appeared south of the zenith in Alexandria. The height of the obelisk was
30 meters and the length of its shadow was 3.79 m long. The Sun was not as high
in the sky in Alexandria as in Syene at the same time of day and the same time of
year. This is in fact a measurement of the curvature of the Earth's sphere between
Syene and Alexandria. Once the distance of this small arc is known, the
circumference of the entire Earth can be determined. To this end (Astronomers
having considerably more political clout in those days!) he ordered some soldiers
to march off the distance between Alexandria and Syene, a distance of 5000 stadia
(thought today to be equivalent to about 500 miles).
Calculate the circumference of the Earth. Find the diameter of the Earth. How
does this value compare to the value in Fundamentals of Astrodynamics?
2. Develop the 2-dimensional (x,y) equation of motion (EOM) for a charged particle
traveling between two charged plates. The force is given by the Lorentz Force



Equation as F  qE where q is the elemental charge (a constant) and E is the

electric field generated between the charged plates. The magnitude of the E
P
vector is E 
where ∆P is the potential difference between the plates and h is
h
the plate separation distance. Assume V and h are constants (given) for this
problem. The direction of the electric field is from the high potential to ground as
shown in the figure below. The charged particle enters the charged plates at a
distance of h/2 above the bottom plate as shown.
Ptop=P
vo
Pbottom=0

E
h