Expert: Philip A. Stahl - 2/1/2009

Question
This is home work. The question is what is the speed of a spacecraft in circular orbit at an altitude of 500, 5000 50000 km. The first answer is 7610, the second is 5900, but I just quessed around and do not understand the math or which process I use with only the altitude . The second part of the question for each is how does the centripteal acceleration compare with the gravitational acceleration. I have none of these answers. I am looking for a direction not for you to do the work.

Answer
Hello,

When doing any sort of physics problem (or astrophysics!) the first priority is to write down all your data - AND ancillary-auxiliary data. (That is, the data not explicitly provided but which is assumed that you know in order to work the problem)

Let's see how this applies here, for the first part of your problem.

You want to find the velocity of the spacecraft (v) so:

v = ?

You want to find it at altitudes (h) of:

a) 500 km

b) 5000 km

c) 50,000 km

The auxiliary data needed are as follows:

G (gravitational constant) = 6.7 x 10^-11 N-m^2/ kg^2

Earth radius R_E = 6.4 x 10^ 6 m

Earth's mass M(E) =   6.0 x 10^24 kg

Now, for any problem involving centripetal force (F_c = mv^2/ r) and force of  gravitation (F_G) - which will apply to all cases of planets orbiting the Sun (e.g. to find speeds of such) OR satellite orbiting Earth, one will use the fact that the gravitational force equals the centripetal force of the orbiting object, whatever it is.

Thus:

F_c  =  F_G

or,

mv^2/ R   =  GMm/R^2

since the mass of the satellite or spacecraft (m) cancels both sides:

v^2/ R  =  GM/R^2

Now, this needs to be adjusted for the situation.

The velocity v, will be that for the spacecraft IF R is modified such that:

R =   R_E  + h   (radius of Earth + altitude)

In which case (taking case (a)):

R =  6400 km +   500 km = 6900 km =  6.9 x 10^6 m

Note you need to convert distances to meters in order to obtain v in m/s!

Similarly, for this to work, M in the above eqn. requires M = M(E) the mass of Earth

Then, using basic algebra (to solve for velocity v):

v  = SQRT [GM(E) / R]  =   (GM(E) /R)^1/2

This yields:

v = 7.88 x 10^3 = 7880 m/s

so you can see your first ans. is off.

As you can gather, the other answers (for altitudes 5000 km, 50,000 km) will be modified as you change the values for R in the workings. (Since h will be different each time, thus R = R_E + h will change)

Using this template as a basis you should be able to work out the other two answers and obtain the correct results, and also be able to apply this method to any other similar problems - including (say on a test) being asked to find the speed of the Jovian satellite Io around Jupiter given its distance!)