Physics - orbits

Here is the question I am stuck on. The answer is 7.

An artificial satellite is in a circular orbit around a moon at radius R = ar.
r is the radius of the moon itself.

The velocity of the satellite is then halved. The satellite's orbit is now one that just grazes the moons surface.

Use keplars first law to infer the shape and orientation of the orbit (done this - its obviously an ellipse with maximum radius ar and minimum radius r).

Find the value of a.

I'm thinking you need to use conservation of energy considering the points where radius = r and radius = ar for the elliptical orbit but cant get the right answer.

Reply 1

elpaw

have you tried conservation of angular momentum too?

Reply 2

Lagz

yes thats how i got my incorrect answer, using the invariance of
dA/dt = L/2m (i think thats right reciting from memory now). I used this to find v2 (the velocity when radius = r) in terms of v. Then substituted into a conservation of energy equation to eliminate everything else.

Got wrong answer though

Reply 3

Lagz

anyone?

Reply 4

4Ed

Lagz

anyone?

if you could just remind me of the relevant Keplar laws again...
i did something like this last year, but i've forgotten all the prereq. laws...

Reply 5

Lagz

I've done it now. Will post solution up in a sec incase any1 is interested.

Reply 6

Lagz

Solution:

Initially the satellite is orbiting at a constant distance R=ar from the moon.

You can apply: mv^2/R = GMm/R^2 => v = SQRT[GM/R] = SQRT[GM/ar]

Now using the second orbit and the fact that the total energy of the satellite is constant:

LHS is when radius = ar. RHS is when radius = r (when its grazing the moon). v2 is its velocity when radius = r

m(v/2)^2/2 - GMm/ar = m(v2)^2/2 - GMm/r (1)

Now the angular momentum L is invarient. |L| = mrv at the two points being used.

|L| is invarient => v2 = va/2

Now this is substituted into (1). Then v = SQRT[GM/ar] is substituted in, everything cancalls leaving a = 7.

(i havent checked that what i have written is the same as what I have written down so if it doesnt all work sorry :biggrin:

)

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