Is the Orbital Energy of a Comet Elliptic, Parabolic, or Hyperbolic?

[student1938]

A comet is first observed at a distance of 1/3 AU from the Sun and traveling at a speed of twice the Earth' s orbital speed. Assuming that the Earth' s orbit is circular, argue from an energy viewpoint whether the comet' s orbit is elliptic, parabolic or hyperbolic.

Now, I know the eccentricity equation but am genuinely lost here...any feedback would be really nice please!

 

[student1938]

Any advice would be great, guys...really I need help.

 

[turin]

What's an AU?

What's the "eccentricity equation?" There are two regions for eccentricity. The values less than 1 are bounded below by the circular eccentricity, and the two regions are divided by the parabolic eccentricity.

Just calculate the total energy of the comet (which remains constant if gravity is assumed to be a conservative force) and then you can infer from the total energy what the eccentricity is, for instance, by looking here:
http://scienceworld.wolfram.com/physics/Eccentricity.html

 

[Janus]

I'm assuming that you know how to calculate the total energy of the Comet (KE+GPE).

If the total energy is negative, the orbit is elliptic.

If the total energy is zero, the orbit is parabolic.

And if the total energy if positive, the orbit is hyperbolic.

 

[enigma]

KE - GPE, isn't it?

 

[Janus]

Originally posted by enigma
KE - GPE, isn't it?

Remember, GPE is a negative term by itself, Thus in the equation you are adding a negative term.

 

[student1938]

Assuming that the Earth' s orbit is circular, argue from an energy viewpoint whether the comet' s orbit is elliptic, parabolic or hyperbolic.


This is the most confusing statement for me.. do I need to use the fact that the Earth' s orbit is circular(except for using this fact to calculate the circular ORBIT speed?)

Can someone clarify this point for me?

 

[enigma]

An orbit is circular only if it has the circular velocity:

$$V = \sqrt{\frac{GM}{r}}$$

An orbit is parabolic if it has the escape velocity (parabolic is the first type of orbit which won't come back to the same point).

$$V_{esc} = \sqrt{2}V_{circ}$$

An orbit is hyperbolic if it has more than the escape velocity

An orbit is elliptical for any other speed.

 

[student1938]

So I am supposed to use twice the circular velocity as per what the question says right?

Assuming that this is the case, then ,E = KE + PE
E = (1/2)*mass of comet*(2*circular velocity)^2 + -G*mass of spacecraft *mass of Earth/((1/3)*AU).. IS this the way it is supposed to go?

 

[enigma]

Why not do this:

$$V_{circ, Earth} = \sqrt{\frac{MG}{r_{Earth}}}$$

$$V_{circ, comet} = \sqrt{\frac{MG}{r_{comet}}}$$

$$V_{circ, comet} = \sqrt{\frac{MG}{\frac{1}{3}r_{Earth}}}$$

$$V_{circ, comet} = \sqrt{3}V_{circ, Earth}$$

Since you know the actual speed of the comet, you can figure out what its speed relative to the circular speed. That'll give you the type of orbit.

 

[student1938]

Thanks for the tip so now I can figure out the circular speed of the Earth...that gives me the Orbital speed of the comet and so I can then get the energy of the comet and then determine the type of orbit..this is the way that I see it...is it correct

 

[enigma]

Yes, that's one way to do it.

You don't need to actually calculate the exact numbers if you don't want to... you can just leave everything in symbolic notation. They didn't ask for specific numbers; just the type of orbit, and that can be determined by relative speeds.

 

[student1938]

Thanks for the help...really eye opening