Thrust needed for Orbit of a satellite

[Kalus]

Homework Statement

A satellite is fitted with a station keeping rocket capable of producing thrust of 500N. Its speed is 500Km/hr. Mission control wishes to increase the radius of orbit by a factor of 0.5. For how long should the rocket be fired to achieve this? (The weight of the satellite and rocket is 200kg, neglect the weight of fuel used during the burn)

Homework Equations

F= Gm1m2/r^2 (although m2 is not known)

F/m1 = Gm2/r^2

a= Gm2/r^2

a= V^2/r

so,

V^2/r = Gm2/r^2

V= root(Gm2/r)

also

m1v1 + F*t = m1v2

The Attempt at a Solution

My idea was to try and find the velocity of orbit at 1.5r, but I am not sure how i can do that without using the mass of the body that the rocket is orbiting about. After that my idea was to use the above momentum formula and rearrange for t? Its really the calculation of the new velocity that I am stuck on.

 

[D H]

The question, as written, is a bit ill-posed. Increase the radius of an orbit (implying circular orbits) requires two burns. The first burn places the vehicle in an elliptical orbit that brings the vehicle to the desired altitude. At some later time the vehicle will be at the desired altitude. The second burn takes place then and circularizes the orbit. The most efficient way of transferring from one circular orbit to another is the Hohmann transfer.

You don't need to know the mass of the planet.

 

[Kalus]

Thanks for your input. We haven't been taught about the Hohmann transfer, so it is unlikely i need theory from it. The question assumes just one burn.

I've realized that from the equations I've listed below i can form:

V1^2 * r = V2^2 * 1.5r = G*m2

However, this suggests that the speed in the second case with a bigger radius will be higher? I thought it was the opposite of this?

 

[D H]

That suggests the speed in the second case is lower. Whenever you have a relationship of the form x^a*y^b=constant, with a and b > 0, then increasing x means you have to decrease y.

Simply decreasing the vehicle's speed from the initial v₁ to v₂ will not achieve the desired goal. It will do just the opposite: It will send the vehicle into a lower orbit (more specifically, an elliptical orbit with a lower semi-major axis). A single burn of any sort will not achieve the desired goal.

Could you write the problem as stated in your text?

 

[Kalus]

The way i have stated the problem is exactally how it is written in the text. The only other information given is 5 possible answers (its from a multiple choice past exam paper). It may assume a simplified situation (e.g. no elliptical orbits and one firing).

If we were to assume it could be done, using the parameters i listed in my first post how could you solve to find a V2 with the new radius 1.5r?

Many thanks, Andy

 

[D H]

What is the difference in mechanical energy (kinetic plus potential) for an object in a circular orbit with radius r versus that for an object in a circular orbit with radius 1.5*r? Whatever that difference is, you will have to supply at least this much energy in terms of delta-V.