Calculating Speed of Asteroid at its Perihelion

[FaraDazed]

This is not part of my coursework; I am preparing for an exam and this is a question from a past paper to which answers are not given.

1. Homework Statement
An asteroid orbiting the sun has its perihelion at 1AU and its aphelion at 3AU. Calculate the speed of the asteroid at its Perihelion.

Homework Equations

Kepler s
Universal Gravitation
UCM ?

The Attempt at a Solution

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I am a bit stuck on this. First I calculated its semi-major axis as I assume it will be needed,

$a=\frac{r_a + r_b}{2}=\frac{(1.49 \times 10^{11})+(3 \times 1.49 \times10^{11}}{2} = 2.98 \times 10^{11} m$

Could I use the uniform circular motion equation $F_c= \frac{mv^2}{r}$ in this situation? I ask as its an a moderately eccentric ellipse.

I was thinking of equating that to the force of gravity to get $v$, but if I could use it in this situation, what would I use as $r$, the semi-major axis, or 1AU?

If I cannot use that UCM equation then I am more stuck than I realized, so would appreciate any help.

Thanks,

 

[TSny]

Since the motion is not UCM, then you would need to justify using the UCM equation with the semi-major axis. I'm pretty sure it won't lead to the right result.

Can you think of a couple of quantities that are conserved in the orbital motion?

What units are you using for distance? The 10¹⁵ factors look strange to me.

 

[FaraDazed]

Since the motion is not UCM, then you would need to justify using the UCM equation with the semi-major axis. I'm pretty sure it won't lead to the right result.

Can you think of a couple of quantities that are conserved in the orbital motion?

What units are you using for distance? The 10¹⁵ factors look strange to me.

Sorry, I remembered incorrectly, it is 10¹¹ in metres.

Angular momentum? But I cannot see how I can work with that, not knowing the asteroids shape etc.

 

[TSny]

Orbital angular momentum is conserved. Treat the asteroid as a point particle.

You will need a second conserved quantity.

 

[FaraDazed]

Orbital angular momentum is conserved. Treat the asteroid as a point particle.

You will need a second conserved quantity.

Ok thanks. Have not done many problems in angular momentum, could I use the Earth?

EDIT: Typo in op, the aphelion is 3AU, not 2.

 

[TSny]

The Earth is not relevant to this question. You just have the asteroid orbiting the sun.

See if this link helps refresh your memory concerning orbital angular momentum: http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

 

[gneill]

Hi FaraDazed. If I might offer a hint: what do you know about the relationship between the semi-major axis of an orbit and its specific mechanical energy?

 

[FaraDazed]

The Earth is not relevant to this question. You just have the asteroid orbiting the sun.

See if this link helps refresh your memory concerning orbital angular momentum: http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

Thanks for the link. I didn't think I could, I just couldn't think what else, as it's not like it has any moons (the only question i did before with regarding the Earth Moon system.

 

[FaraDazed]

Hi FaraDazed. If I might offer a hint: what do you know about the relationship between the semi-major axis of an orbit and its specific mechanical energy?

Nothing until now, did not know there was a link. But after doing a quick google search I found this http://en.wikipedia.org/wiki/Vis-viva_equation , thanks!