kinetic and potential problem help please

[rinsterman12]

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because of the periodic nature of the orbit a simple relation between average ke is <k> of the companion star<u>. In particular, <k> =<u/2>.
For a circular orbit, the potential energy of the companion star is constant throughout the orbit. If the radius of the orbit doubles what happens to velocity?

To me u decrease to u/2 which equals to 1/2mv^2, so u/2 = 1/2mv^2. From this express can you infer that kinetic energy decreases by half? Also that the velocity would equal 1/ square of 2.

why would the potential not be u/4 since potential is halfed and if you plug in u/2/2 then you get a fourth of the potential?

 

[lorenzomicron]

The question is unclear because of this part : ( For a circular orbit, the potential energy of the companion star is constant throughout the orbit. If the radius of the orbit doubles what happens to velocity?) which radius doubles? which velocity are we considering?

 

[rinsterman12]

I am having a misconception about proportionality. What I really want to know is how does doubling the radius in the formula, U= - gmM/r. How does that half the kinetic energy in the formula <k> = - <U/2>? To me that would be -0.5 from the potential then -(-.5/2) = a positive number. Please help It would seem that it should be .25.

 

[lorenzomicron]

Let's see, double the radius will cause U to become U/2. then, then, the coupled kinetic will become K=U/4, the new k is 1/2 of the old k

however, the question asked for what happens to velocity.

k=1/2 mv^2, new k = 1/2 old k= 1/2 mv^2

k new = mv^2, this means the velocity is doubled.

that right?

 

[rinsterman12]

how is new k 1/2 the old k? Does it have to do with the negative in the formula? If old k = - u/2 then new k = - u/4? Set this up as a ratio and that would get 2 old / new, right? Or the new one is 1/2 the old one?

 

[lorenzomicron]

the negative formula or anything doesnt matter. just looking at the change caused by r becoming 2r

best way to do this is to set this up as a ratio

work it out u get

U new/ U old = 1/2

this means Knew/Kold = 1/4

this means (1/2m*vnew^2)/(1/2m*vold^2) = 1/4
solving this gives vnew/vold = 2