Many satellites are moving in a circle in the earth's equatorial plane. They are at such a height above the earth's surface that they always remain above the same point.
(a) Find the altitude of these satellites above the earth's surface. (b) Explain, with a sketch, why the radio signals from these satellites cannot directly reach receivers on earth that are north of 81.3 N latitude.

Question

30-11-2022
Geography

Answered

Many satellites are moving in a circle in the earth's equatorial plane. They are at such a height above the earth's surface that they always remain above the same point.
(a) Find the altitude of these satellites above the earth's surface. (b) Explain, with a sketch, why the radio signals from these satellites cannot directly reach receivers on earth that are north of 81.3 N latitude.

Answer :

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syadzayasmine syadzayasmine · Answer 1 · 2022-12-02T11:23:51-05:00

The altitude of these satellites above the earth's surface is 3,588 10⁷ m, and the radio signals from these satellites cannot directly reach receivers on earth that are north of 81.3 N latitude because by the curve of the planet; consequently, the signals cannot reach these places

to find the altitude of these satellites above the earth's surface requires the use of the law of universal gravitation and Newton's second law.

F = ma

G m M / r² = m a (1)

centripetal acceleration

a = v²/r

Why is the velocity factor (velocity) constant?

v = d/t

A complete orbit with a time (T) called the period circumnavigates the distance.

d = 2πr

v = 2πr/T

a = (4π²r²/T²)/r

insert in expression 1

GM/r² = 4π²r/T²

GM/r³ = 4π²/T²

r = ∛ GM T² / 4π²

R is the distance from the center of the earth which is the distance from the surface of the earth

R = Re + h

Re + h = ∛ GM T²/4π²

h = ∛(G M T²/4π²) - Re

Convert to SI units

T = 1 day (24 hours / 1 day) (3600 seconds / 1 hour)

T = 86400 seconds

let's do the math

h = ∛ (6.67 10⁻¹¹ 5.98 10²⁴ (8.6400 10⁴)² / 4π²) - 6.37 10⁶

h = ∛ (75.42 10²¹) - 6.37 10⁶

h = 4.225 x 10⁷ - 0.637 x 10⁷

h = 3,588 10⁷m

b) In the appendix we find that for very large latitudes the straight path from the satellite to the earth is interrupted by the curvature of the planet. As a result, the signal cannot reach these locations