Total energy of satellite in orbit

[cyt91]

Homework Statement

The weight of a satellite in a circular orbit round the Earth is half of its weight on the surface of the Earth. If the mass of the satellite is 800kg, determine
(i)the altitude of the satellite
(ii)the speed of the satellite in the orbit
(iii)the minimum energy required to move the satellite from its orbit to outer space.

Radius of Earth = 6.4 x 10^6 m
Mass of Earth = 6.0 x10^24 kg

Homework Equations

Gravitational force = -GMm/(r^2)
W=mg
Gravitational potential energy = -GMm/r
Kinetic energy of satellite = GMm/(2r)

The Attempt at a Solution

(i) GMm/(r^2)=mg
GMm/(r'^2)=0.5mg
Solving, r'=(2^0.5)r
Altitude,h=r'-r
=(2^0.5-1)(6.4 x 10^6)
= 2.7 x 10^6 m

(ii) GMm/(r^2)= (mv^2)/r
v=(GM/(r)^0.5
Solving using r=2^0.5 x 6.4 x 10^6 m
v = 6.7 x 10^3 m/s

(iii) Total energy of satellite in orbit is
T=U+K
When removed to outer space,energy of satellite should be U=0 and K=0 since we want to find the minimum energy required.
Thus,
Energy required,E= 0 - (U+K)
E = 0 -(-GMm/r-GMm/(2r))
= GMm/(2r)
Solving using r= (2^0.5) x 6.4 x 10^6 m and m= 800kg
E=1.8 x 10^7 J

Please check my working,especially the third part of the question. When calculating Minimum energy required to remove the satellite to outer space, we should take into account the kinetic energy of the satellite shouldn't we?

 

[fuzzywolf]

I think you've got it essentially right. When calculating minimum escape energy, the idea is to include the kinetic energy due to tangential velocity and then apply force in the tangential direction. This takes a lot less energy than directing your force radially and moving out to infinity -- half as much, in fact, I think.

Your algebra looks solid.

 

[cyt91]

I think you've got it essentially right. When calculating minimum escape energy, the idea is to include the kinetic energy due to tangential velocity and then apply force in the tangential direction. This takes a lot less energy than directing your force radially and moving out to infinity -- half as much, in fact, I think.

Your algebra looks solid.

Thanks. Then the answer provided by my book is incorrect. They neglected the initial kinetic energy of the satellite. Thanks for checking my work.