MAE 4410: Fundamentals of Astronautics

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 E h

Pbottom=0