Question

A satellite of mass 180 kg is placed into Earth orbit at a height of 350 km above the surface. (a) Assuming a circular orbit, how long does the satellite take to complete one orbit? h (b) What is the satellite's speed? m/s (c) Starting from the satellite on the Earth's surface, what is the minimum energy input necessary to place this satellite in orbit? Ignore air resistance but include the effect of the planet's daily rotation. J

Answer 1

the radius of the orbit r = R_E + h

= 6.37 X 10⁶ + 0.35 X 10⁶

= 6720000 m

a )

using

T² = ( 4² / G M ) r³

T² = 4 X 3.14² X 6720000³ / 6.67 X 10^-11 X 5.97 X 10²⁴

T² = 30055706.53

T = 5482.30 sec

b )

V_f = ( GM / r )^1/2

= ( 6.67 X 10^-11 X 5.97 X 10²⁴ / 6720000 )^1/2

V_f = 7697.77 m/s

c )

V_i = 2 R_E / 24 hrs

= 2 X 3.14 X 6.37 X 10⁶ / 86400

= 463 m/s

E = 1/2 m ( V_f² - V₁² ) + G M m ( 1/R_E - 1/ r )

E = 0.5 X 180 X ( 7697.77² - 463² ) + 6.67 X 10^-11 X 5.97 X 10²⁴ X 180 X ( 1/6.37 X 10⁶ - 1 / 6720000 )

E = 5.31 X 10⁹ + 5.86 X 10⁹

E = 1.117 X 10¹⁰ J