Gravitational Potential Energy in orbit

[ual8658]

This pertains to a homework question but I get the concept of PE or U = -GmM/a for an elliptical orbit. I also understand the derivation of the total energy of an object in an elliptical orbit as E = -GmM/2a. However, I have a homework question that asks for the ratio of an object's kinetic energy to total energy in orbit, and the problem states to use the relationship of K/E = 1 - U/E. However, wouldn't U/E be greater than 1 since E has a denominator of 2a while U has a denominator that is always smaller than 2a? This would force U to be greater than E, which would mean U/E is greater than 1, which leads to a K/E greater than 1. How does this make sense?

 

[Doc Al]

and the problem states to use the relationship of K/E = 1 - U/E.

This is simply E = K + U rearranged.

 

[Doc Al]

This would force U to be greater than E, which would mean U/E is greater than 1,

That's true. U/E = 2.

which leads to a K/E greater than 1.

That doesn't follow. K/E = -1.

 

[ual8658]

That's true. U/E = 2.That doesn't follow. K/E = -1.

Ok I'm used to having ratios that are decimals rather than integer values. So it's acceptable in this case to have 2 and -1 be ratio values?

 

[Doc Al]

Ok I'm used to having ratios that are decimals rather than integer values.

You can express an integer as a decimal if you like.

So it's acceptable in this case to have 2 and -1 be ratio values?

Why not?

 

[ual8658]

You can express an integer as a decimal if you like.Why not?

From using the simple relationship mgh and .5mv^2 when we define total energy, U or K has always been less than the total E. But with this I guess the relationship is opposite since a U value of 0 would indicate infinite distance? Would this also mean that a planet with a larger semi-major axis has more total energy since its E value would be closer to 0?

 

[Doc Al]

From using the simple relationship mgh and .5mv^2 when we define total energy, U or K has always been less than the total E.

Note that the reference point where U = 0 (where h = 0) is arbitrary, so you can get U < 0 there as well.

But with this I guess the relationship is opposite since a U value of 0 would indicate infinite distance?

When dealing with gravity between planets and such, it is most convenient to set U = 0 when they are infinitely far apart. That is the basis of the formulas you quoted earlier.

Would this also mean that a planet with a larger semi-major axis has more total energy since its E value would be closer to 0?

Yes.

 

[ual8658]

Note that the reference point where U = 0 (where h = 0) is arbitrary, so you can get U < 0 there as well.When dealing with gravity between planets and such, it is most convenient to set U = 0 when they are infinitely far apart. That is the basis of the formulas you quoted earlier.Yes.

Ok thank you. You've just cleared up a ton of confusion!