Question

A 150-kg satellite is in circular orbit of radius 7.3 Mm around Earth. Determine:(a) potential, kinetic, and total mechanical energies.(b) the orbital speed.(c) the escape velocity from this altitude.

Answer 1

a)
Potential energy, PE = - GMm/R
Where G is the universal gravitational constant, M is the mass of earth, m is the mass of satellite and R is the orbital radius of the satellite.
PE = - [6.674 x 10^-11 x 5.972 x 10²⁴ x 150] / (7.3 x 10⁶)
= - 8.19 x 10⁹ J

b)
Force between the satellite and earth, F = GMm/R² = mv²/R
Where mv²/R is the centripetal force
GMm/R = mv²           [Multiplying with R on both the sides]
Kinetic energy = 1/2 mv² = 1/2 GMm/R               [Multiplying with 1/2 on both the sides]
= 1/2 x PE
= 1/2 x 8.19 x 10⁹ J
= 4.095 x 10⁹ J

c)
Total mechanical energy = KE + PE
= 1/2 GMm/R + [-GMm/R]
= - 1/2 GMm/R
= - 4.095 x 10⁹ J

d)
From part b, 1/2 mv² = 4.095 x 10⁹ J
v² = ( 2 x 4.095 x 10⁹) /150
v = SQRT[( 2 x 4.095 x 10⁹) /150]
= 7.39 x 10³ m/s

e)
When the satellite attains escape velocity, its total energy will be zero.
KE + PE = 0
1/2 mv² - GMm/R = 0
1/2 mv² = GMm/R
= 8.19 x 10⁹                           [From part a]
v² = ( 2 x 8.19 x 10⁹) /150
v = SQRT[( 2 x 8.19 x 10⁹) /150]
= 10.45 x 10³ m/s